text:
\section{Introduction}
In evolution problems for interfaces, the occurrence of topology changes and the associated geometric singularities generally limits the applicability of classical solution concepts to a finite time horizon, depending on the initial data.
The evolution beyond topology changes can only be described in the framework of suitably weakened solution concepts. However, weak concepts may in general suffer from an (unphysical) loss of uniqueness of solutions: For example, in the framework of Brakke solutions \cite{Brakke} to mean curvature flow (MCF), the interface may suddenly disappear at any time (see Figure~\ref{FigureBrakke} for an illustration). In particular, Brakke solutions fail to be unique, even prior to the onset of geometric singularities in the classical solution.
With the exception of evolution equations subject to a comparison principle such as two-phase mean curvature flow \cite{ChenGigaGoto,EvansSpruck}, only few positive results on uniqueness of weak solutions for interface evolution problems are known.

In the present work, we establish a weak-strong uniqueness principle for distributional solutions (in the framework of finite perimeter sets, a solution concept also known as ``$\BV$ solutions'') to multiphase mean curvature flow: As long as a strong solution to planar multiphase mean curvature flow -- in the sense of an evolution of smooth curves meeting at triple junctions at an angle of $120^\circ$ -- exists, any
distributional
solution starting from the same initial conditions must coincide with it. Note that for regular initial data, strong solutions are known to exist until a topology change in the network of evolving curves occurs, see for instance \cite{MantegazzaNovagaPludaSchulze}. In particular, our result establishes uniqueness of
distributional
solutions to planar multiphase mean curvature flow in the absence of topology changes.

The key insight in our present work is the observation that in analogy to the notion of calibrations for minimizers of the surface energy functional, one may develop a notion of calibrations for its gradient flow.
Just like classical calibrations carry information on the global structure of the energy landscape -- namely, a global lower bound for the energy -- , ``gradient flow calibrations'' carry information on the local structure of the energy landscape near a partition evolving by mean curvature: The existence of a gradient flow calibration implies that the path of steepest descent in the energy landscape of the surface energy functional is unique and stable with respect to perturbations of the initial condition.\footnote{While in the present work ``path of steepest descent'' is to be understood as ``$\BV$ solution to multiphase mean curvature flow'', we will give a rigorous statement of this notion at the level of the energy functional in a future work.}

We implement this strategy in general ambient dimension $d\geq 2$ by proving that the existence of a gradient flow calibration implies an inclusion principle for BV solutions to multiphase mean curvature flow ensuring that they are contained in the calibrated flow.
This essentially reduces proving the desired weak-strong uniqueness to the construction of such a gradient flow calibration for strong solutions, which we provide in the planar case $d=2$.
However, we would like to emphasize that conceptually the approach carries over to multiple dimensions. In particular, with the techniques used in the present paper it is for example possible to calibrate the smooth evolution of a double bubble; the adaptation of our arguments will be elaborated on in the upcoming note \cite{FischerHenselLauxSimonMultiD}. However, as soon as quadruple junctions (as they typically occur in three spatial dimensions) are present in the initial data, an additional construction is needed; nevertheless, we expect the principles of our present construction to guide the construction also in this situation.

\begin{figure}
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\end{tikzpicture}
\caption{A partition of a planar domain by a network of smooth curves meeting at triple junctions at angles of 120$^\circ$, corresponding to the typical situation in multiphase mean curvature flow with equal surface energies.\label{FigureGrains}}
\end{figure}

\subsection{Multiphase mean curvature flow}

Mathematically, mean curvature flow is one of the most studied geometric evolution equations. Being the gradient flow of the area functional with respect to the $L^2(S_t)$ distance, it constitutes the perhaps most natural area-reducing flow for submanifolds.
Its multiphase variant may be seen as the simplest case of mean curvature flow for a non-smooth surface, allowing for ``branching'' of the surface (see e.\,g.\ Figure~\ref{FigureGrains}).

Multiphase mean curvature flow also is an important phenomenological model for the motion of grain boundaries in polycrystals (``grains'' being the domains in a polycrystal with a single crystallographic orientation): Their evolution may be approximated as the gradient flow of the surface energy between the different grains, see for instance the seminal work of Mullins \cite{Mullins}.
While in principle the motion of grain boundaries is governed by anisotropic mean curvature flow or even more complex evolution equations \cite{SrolovitzUnified,SrolovitzEtAl}, isotropic multiphase mean curvature flow may be viewed as an important model case for these
equations.
For recent developments in anisotropic and crystalline curvature flows, we refer to Caselles and Chambolle \cite{CasellesChambolle} and Chambolle, Morini, and Ponsiglione \cite{ChambolleMoriniPonsiglione}.

The existence theory for solutions to multiphase mean curvature flow is
quite well-developed:
Classical solutions to planar multiphase mean curvature flow are known to exist (and to be unique) for short times, see Bronsard and Reitich \cite{BronsardReitich}.
For initial configurations close to an equilibrium state, classical solutions exist even globally in time, see Kinderlehrer and Liu \cite{KinderlehrerLiu}. In the higher-dimensional case, Depner, Garcke, and Kohsaka \cite{DepnerGarckeKohsaka} have shown the local-in-time existence of classical solutions for the evolution of a double bubble.
In principle, Brakke's concept of varifold solutions \cite{Brakke} is applicable to multiphase mean curvature flow. However, it suffers from the well-known shortcoming of exhibiting a drastic and unphysical failure of uniqueness of solutions \cite{Brakke} as mentioned above; see Figure~\ref{FigureBrakke} for an illustration.

\begin{figure}
\begin{tikzpicture}[xscale=0.7,yscale=0.7]
\draw (-2,0) ellipse (1.0 and 1.0);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.5,yscale=0.7]
\draw[white] (0,-1) -- (0,1);
\draw[very thick,->] (-0.4,0) -- (0.4,0);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.7,yscale=0.7]
\draw[white] (-3,-1) -- (-1,1);
\draw (-2,0) ellipse (0.8 and 0.8);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.5,yscale=0.7]
\draw[white] (0,-1) -- (0,1);
\draw[very thick,->] (-0.4,0) -- (0.4,0);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.7,yscale=0.7]
\draw[white] (-3,-1) -- (-1,1);
\draw (-2,0) ellipse (0.6 and 0.6);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.5,yscale=0.7]
\draw[white] (0,-1) -- (0,1);
\draw[very thick,->] (-0.4,0) -- (0.4,0);
\end{tikzpicture}
\begin{tikzpicture}[xscale=0.7,yscale=0.7]
\draw[white] (-3,-1) -- (-1,1);
\draw[fill=black] (-2,0) circle (0.03);
\end{tikzpicture}
\\~\\
\begin{tikzpicture}[xscale=0.7,yscale=0.7]
\draw (-2,0) ellipse (1.0 and 1.0);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.5,yscale=0.7]
\draw[white] (0,-1) -- (0,1);
\draw[very thick,->] (-0.4,0) -- (0.4,0);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.7,yscale=0.7]
\draw[white] (-3,-1) -- (-1,1);
\draw (-2,0) ellipse (0.8 and 0.8);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.5,yscale=0.7]
\draw[white] (0,-1) -- (0,1);
\draw[very thick,->] (-0.4,0) -- (0.4,0);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.7,yscale=0.7]
\draw[white] (-3,-1) -- (-1,1);
\end{tikzpicture}
~
\begin{tikzpicture}[xscale=0.5,yscale=0.7]
\draw[white] (0,-1) -- (0,1);
\draw[very thick,->] (-0.4,0) -- (0.4,0);
\end{tikzpicture}
\begin{tikzpicture}[xscale=0.7,yscale=0.7]
\draw[white] (-3,-1) -- (-1,1);
\end{tikzpicture}
\caption{Top: An initially circular interface evolving by mean curvature flow. In finite time the interface shrinks to a point and disappears, giving rise to a geometric singularity and a topology change. Bottom: In Brakke solutions to mean curvature flow, the interface may suddenly disappear at any time, leading to a drastic failure of uniqueness of solutions.\label{FigureBrakke}}
\end{figure}

The existence of classical solutions to planar multiphase mean curvature flow up to finitely many singular times -- a solution concept that we will refer to as ``classical solutions with restarting'' -- has been established by Manteganzza, Novaga, Pluda, and Schulze \cite{MantegazzaNovagaPludaSchulze} under the assumption that certain types of singularities do not accumulate, extending earlier results by Ilmanen, Neves, and Schulze \cite{IlmanenNevesSchulze} and Mantegazza, Novaga, and Tortorelli \cite{MantegazzaNovagaTortorelli}. However, it is not evident how to generalize this notion of solutions to the higher-dimensional case, as it relies on the classification of potential singularities.

In \cite{LauxOtto,LauxOttoBrakke}, a conditional convergence result for an efficient numerical scheme for multiphase mean curvature flow -- the thresholding scheme of Merriman, Bence, and Osher \cite{MerrimanBenceOsher} -- towards $\BV$ solutions of multiphase mean curvature flow has been shown by Otto and the third author, thereby also establishing a conditional existence result for $\BV$ solutions. In \cite{LauxSimon}, a conditional convergence result for the Allen-Cahn approximation for multiphase mean curvature flow towards $\BV$ solutions has been derived by the third and the fourth author. Both results employ an assumption of convergence of the interface area, analogous to the one in Luckhaus-Sturzenhecker \cite{LuckhausSturzenhecker} for the implicit time discretization developed by Luckhaus-Sturzenhecker and Almgren-Taylor-Wang \cite{AlmgrenTaylorWang}.

\subsection{The uniqueness properties of multiphase mean curvature flow}

The uniqueness properties of weak solution concepts for multiphase mean curvature flow have remained essentially unexplored.
For two-phase mean curvature flow, a combination of the level-set formulation by Osher and Sethian \cite{OsherSethian} and Ohta, Jasnow, and Kawasaki \cite{OhtaJasnowKawasaki}, and the concept of viscosity solutions by Crandall and Lions \cite{CrandallLions} facilitates an existence and uniqueness theory for a weak notion of solutions, as shown by Chen, Giga, and Goto \cite{ChenGigaGoto} and Evans and Spruck \cite{EvansSpruck}. While these viscosity solutions to two-phase mean curvature flow are unique, a given level set may ``fatten'' \cite{BarlesSonerSouganidis}, thereby failing to describe an interface and indicating the emergence of a non-unique evolution of the surface. Nevertheless, fattening is known to not occur prior to the first topology change, provided that one starts with a smooth initial surface.
Unfortunately, the absence of a comparison principle for multiphase mean curvature flow a~priori prevents the applicability of these techniques in the multiphase case.

Recently, a notion of solutions for multiphase mean curvature flow has been proposed by Kim and Tonegawa \cite{KimTonegawa} which is likely to exclude the unphysical sudden vanishing of the interface in Brakke solutions.
While we have not yet been able to establish a weak-strong uniqueness principle for their solution concept, it seems likely that our techniques may also contribute to the analysis of the uniqueness properties of this notion of solutions.

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\end{scope}
\begin{scope}[xscale=0.7,yscale=0.7,xshift=3.2cm,yshift=2.5cm]
\draw[fill=BLUE!70!white,draw=black,thick] (0+5.0,0) -- (2+5.0,0) -- (2+5.0,2) -- (0+5.0,2) -- cycle;
\draw[fill=GREEN!70!white,draw=black,thick] (0+5.0,0) -- (2+5.0,0) -- (2+5.0,2) -- cycle;
\draw[fill=PURPLE!70!white,draw=black,thick] (0+5.0,0) -- (0+5.0,1) .. controls (0.45+5.0,1) and (0.85-0.2*0.97+5.0,0.85+0.2*0.258) .. (0.85+5.0,0.85) .. controls (0.85+0.2*0.258+5.0,0.85-0.2*0.97) and (1+5.0,0.45) .. (1+5.0,0) -- cycle;
\draw[fill=RED!70!white,draw=black,thick] (2+5.0,2) -- (2+5.0,1) .. controls (1.55+5.0,1) and (1.15+0.2*0.97+5.0,1.15-0.2*0.258) .. (1.15+5.0,1.15) .. controls (1.15-0.2*0.258+5.0,1.15+0.2*0.97) and (1+5.0,1.55) .. (1+5.0,2) -- cycle;
\draw[thick] (0+5.0,0) -- (2+5.0,0) -- (2+5.0,2) -- (0+5.0,2) -- cycle;
\end{scope}
\end{tikzpicture}
\caption{An example of a nonunique evolution of multiphase mean curvature flow, starting from an initial interface consisting only of smooth curves meeting at an angle of 120$^\circ$.\label{FigureNonuniqueness}}
\end{figure}

The example in Figure~\ref{FigureNonuniqueness} shows that after topology changes, the uniqueness of BV solutions to planar multiphase mean curvature flow may fail.
Note that in contrast to the sudden vanishing of the interface in Brakke solutions, this is a case of \emph{physical} non-uniqueness: The failure of uniqueness is caused by a physically unstable situation -- the symmetric configuration of four perfect squares -- , starting from which infinitesimal perturbations may select either of the two evolutions.
This example also shows that a principle of maximal dissipation of energy may fail to single out a unique evolution.

Our main result -- a uniqueness theorem for $\BV$ solutions to planar multiphase mean curvature flow prior to the first topology change -- is therefore not only the first positive result concerning uniqueness for a weak solution concept to multiphase mean curvature flow, but also optimal for general initial data.
Nevertheless, let us mention that it has been suggested by Ilmanen (see e.\,g.\ \cite{MantegazzaNovagaPludaSchulze}) that the uniqueness properties may be better if one restricts one's attention to \emph{generic} initial data:
For initial data given by a small random perturbation of a fixed multiphase interface, the evolution by mean curvature in the plane is expected to be unique and stable with respect to perturbations for \emph{almost every} perturbation.
The argument in favor of this proposed phenomenon is based on a numerical study classifying the ``stable'' and therefore ``generically occurring'' singularities in planar mean curvature flow \cite{Haettenschweiler}.
Evidence in favor of ``generic'' well-posedness is abundant in two-phase mean curvature flow: For instance, an infinitesimal amount of stochastic noise has been shown to yield selection principles for the evolution, see Dirr, Luckhaus, and Novaga \cite{DirrLuckhausNovaga} and Souganidis and Yip \cite{SouganidisYip}. Furthermore, in the framework of viscosity solutions it is immediate that fattening of level sets must be absent in almost all levels. Finally, a classification of generic singularities has been achieved by Colding and Minicozzi \cite{ColdingMinicozzi2,ColdingMinicozzi}.

\begin{table}
\begin{tabular}{llll}
\emph{Solution concept}
&\emph{Topology changes}
&\emph{Uniqueness prior to}
&\emph{Existence}
\\&&\emph{topology changes}&\emph{theory}
\vspace{2.5mm}
\\
classical solutions
&not possible
&yes \cite{BronsardReitich}
&yes \cite{BronsardReitich}
\vspace{1.5mm}
\\
Brakke solutions
&possible
&no \cite{Brakke}
&yes \cite{Brakke}
\vspace{1.5mm}
\\
Kim-Tonegawa solutions
&possible
&? (likely yes\footnotemark)
&yes \cite{KimTonegawa}
\vspace{1.5mm}
\\
classical solutions with
&possible
&yes \cite{MantegazzaNovagaPludaSchulze}
&yes\footnotemark \cite{MantegazzaNovagaPludaSchulze}
\\
restarting (2D only)
\vspace{1.5mm}
\\
$\BV$ solutions
&possible
&{\bf yes (Theorem\,\ref{MainResult})}
&yes\footnotemark \cite{LauxOtto,LauxSimon}
\vspace{2.5mm}
\end{tabular}
\caption{An overview of solutions concepts for multiphase mean curvature flow.
\label{TableSolutionConcepts}}
\vspace{-5mm}
\end{table}
\footnotetext[2]{Provided that one starts with a multiplicity one interface.}
\footnotetext[3]{Global existence under the assumption that a certain type of singularities does not accumulate.}
\footnotetext[4]{Global existence under an assumption as in Almgren-Taylor-Wang / Luckhaus-Sturzenhecker.}

\subsection{Classical calibrations and gradient flow calibrations}

The key idea for our weak-strong uniqueness result is a gradient-flow analogue of the notion of \emph{calibrations}.
The classical concept of calibrations is an important tool to deduce lower bounds on the interface energy functional for fixed boundary conditions.
Recall that a classical calibration for a candidate minimizer $(\bar \chi_1,\ldots,\bar \chi_P)$ of the interface energy functional (for given boundary conditions and with equal surface tensions) is a collection of vector fields $\xi_i$, $1\leq i\leq N$, subject to the following three properties:
\begin{itemize}
\item It holds that $|\xi_i-\xi_j|\leq 1$ for all $i$ and $j$.
\item The vector fields are solenoidal, i.\,e.,\ $\nabla \cdot \xi_i=0$ for all $i$.
\item On the interface $\partial  \{\bar\chi_i=1\} \cap \partial\{ \bar \chi_j=1\}$ between the phases $i$ and $j$ ($i\neq j$), the vector field $\xi_{i,j}:=\xi_i-\xi_j$ coincides with the outer unit normal vector field of $\partial \{\bar\chi_i=1\}$.
\end{itemize}
The existence of a calibration allows to infer that the partition $(\bar \chi_1,\ldots,\bar \chi_P)$ indeed minimizes the interface energy functional among all possible Caccioppoli partitions, see \cite[Definition 4.16]{AmbrosioFuscoPallara}, of the underlying set $D \subset \Rd[d]$, $d\geq 2$ with the same boundary conditions: For any competitor partition $(\chi_1,\ldots,\chi_{P})$, one may compute using the first two defining conditions of a calibration (with the abbreviation $I_{i,j}:=\partial^*\{\chi_i=1\}\cap \partial^* \{\chi_j=1\}$)
\begin{align*}
E[\chi]&=\frac{1}{2}\sum_{i,j=1,i\neq j}^P \int_{I_{i,j}} 1 \,\mathrm{d}\mathcal{H}^{d-1}
\geq
\frac{1}{2} \sum_{i,j=1,i\neq j}^P \int_{I_{i,j}} (\xi_i-\xi_j)\cdot \frac{\nabla \chi_i}{|\nabla \chi_i|} \,\mathrm{d}|\nabla \chi_i|
\\&
=\sum_{i,j=1,i\neq j}^P \int_{I_{i,j}} \xi_i \cdot \frac{\nabla \chi_i}{|\nabla \chi_i|} \,\mathrm{d}|\nabla \chi_i|
=\sum_{i=1}^P \int_{D} \xi_i \cdot \frac{\nabla \chi_i}{|\nabla \chi_i|} \,\mathrm{d}|\nabla \chi_i|
\\&
=\sum_{i=1}^P \int_{\partial D} \chi_i \vec{n}_{\partial D}\cdot \xi_i \,\mathrm{d}\mathcal{H}^{d-1}.
\end{align*}
The third defining condition for a calibration shows that in the previous computation, equality is in fact achieved for $(\bar \chi_1,\ldots,\bar \chi_{P})$. This proves $E[\chi]\geq E[\bar \chi]$ for all $\chi$ with the same boundary conditions $\chi=\bar \chi$ on $\partial D$.

We recall that a notion of calibrations is also available for free discontinuity problems \cite{AlbertiBouchitteDalMaso}, having important implications for the numerical computation of minimizers \cite{ChambolleCremersPock}. For further applications of the concept of calibrations to anisotropic or nonlocal perimeters, we refer to \cite{AlterCasellesChambolle,CasellesChambolleMollNovaga} and \cite{CabreCalibrations,PagliariHalfspaces}.
\begin{table}
\begin{tabular}{ll}
\emph{Calibrations}
&\emph{Gradient flow calibrations}
\vspace{2mm}
\\
Existence implies global minimality
&Existence implies uniqueness of
\\of surface energy among all partitions
&$\BV$ solutions to gradient flow
\vspace{2mm}
\\
Shortness condition
&Coercivity condition
\\
$|\xi_{i,j}|\leq 1$
&$|\xi_{i,j}|\leq \max\{1-c\dist^2(x,{\bar{I}}_{ij}),0\}$
\vspace{2mm}
\\
Stationary situation
&Advection equation
\\
($\partial_t \xi_i\equiv 0$, $B\equiv 0$)
&$\partial_t \xi_{i,j}+(B\cdot \nabla)\xi_{i,j}+(\nabla B)^{\mathsf{T}} \xi_{i,j}$
\\&$~~~~~~~~~~~~~~~~~~~~~~~~= O(\dist(x,{\bar{I}}_{ij}))$
\vspace{1.5mm}
\\
Vector fields solenoidal
&Motion by mean curvature
\\
$\nabla \cdot \xi_i=0$
&
$\xi_{i,j}\cdot B=-\nabla \cdot \xi_{i,j}+O(\dist(x,{\bar{I}}_{ij}))$~~~~~
\vspace{2.5mm}
\end{tabular}
\caption{A comparison of the concept of calibrations for minimal partitions with the new concept of gradient flow calibrations.
\label{TableCalibrationsGradientFlow}
}
\end{table}

In the present work, we introduce a gradient-flow analogue of the notion of calibrations.
As shown above, the existence of a (classical) calibration ensures that a certain configuration is a global minimizer of the energy functional. In a similar spirit, the existence of a gradient-flow calibration ensures that the path of steepest descent in the energy landscape of the surface energy functional is unique, and moreover that this path is stable with respect to perturbations in the initial condition.
In the case of equal surface tensions, a \emph{gradient flow calibration} for a given classical solution $\bar \chi=(\bar \chi_1,\ldots,\bar \chi_P)$ to multiphase mean curvature flow on $\Rd$ consists of the following objects:
\begin{subequations}
\begin{itemize}
\item A vector field $\xi_{i,j}$ for each pair of phases $1\leq i,j\leq P$, $i\neq j$. Denoting by ${\bar{I}}_{i,j}$ the interface between the phases $i$ and $j$ in the strong solution $\bar \chi$ and by $\bar{ \vec{n}}_{i,j}$ its unit normal vector field pointing from phase $i$ to phase $j$, we require $\xi_{i,j}$ to be an extension of $\bar{\vec{n}}_{i,j}$ subject to the coercivity condition
\begin{align}
\label{LengthBound}
|\xi_{i,j}|\leq \max\{1-c \dist^2(x,{\bar I}_{i,j}),0\}
\end{align}
for some $c>0$.
\item The extended normal vector fields $\xi_{i,j}$ must have the structure $\xi_{i,j}=\xi_j-\xi_i$ for some vector fields $\xi_i$, $1\leq i\leq P$. (This structure is reminiscent of the corresponding condition for classical calibrations and in fact serves a similar purpose, see the explanation preceding \eqref{RelativeEntropyRewritten} below.)
\item A \emph{single} velocity field $B$, which approximately transports \emph{all} extended normal vector fields $\xi_{i,j}$ in the sense
\begin{align}
\label{FirstEq}
\partial_t \xi_{i,j} + (B\cdot \nabla) \xi_{i,j} + (\nabla B)^{\mathsf{T}} \xi_{i,j} = O(\dist(x,{\bar{I}}_{i,j})).
\end{align}
Furthermore, the length of the extended normal vector fields is transported to higher accuracy in the sense
\begin{align}
\partial_t |\xi_{i,j}|^2 + (B\cdot \nabla) |\xi_{i,j}|^2 = O(\dist^2(x,{\bar{I}}_{i,j})).
\end{align}
\item Near the interfaces ${\bar{I}}_{i,j}$ of the strong solution, the normal velocity $\xi_{i,j}\cdot B$ is given by the mean curvature of ${\bar{I}}_{i,j}$ in the sense
\begin{align}
\label{ThirdEq}
\xi_{i,j} \cdot B = -\nabla \cdot \xi_{i,j} + O(\dist(x,{\bar{I}}_{i,j})).
\end{align}
Note that on the interface ${\bar{I}}_{i,j}$, the expression $-\nabla \cdot \xi_{i,j}$ is exactly equal to its mean curvature.
\end{itemize}
\end{subequations}

If a gradient flow calibration exists, we may introduce a measure for the difference between any $\BV$ solution $\chi=(\chi_1,\ldots,\chi_P)$ to multiphase mean curvature flow and the strong solution $\bar \chi$ by defining (with $I_{i,j}:=\partial^* \{\chi_i=1\} \cap \partial^* \{\chi_j=1\}$ denoting the interface between phases $i$ and $j$ and with $\vec{n}_{i,j}$ being its normal pointing from phase $i$ to phase $j$)
\begin{align}
\label{ErrorFunctional}
E[\chi|\xi]:=
\frac{1}{2}\sum_{i,j=1,i\neq j}^{P} \int_{I_{i,j}} 1 - \xi_{i,j} \cdot \vec{n}_{i,j} \,\mathrm{d}\mathcal{H}^{d-1}.
\end{align}
Note that the condition \eqref{LengthBound} then precisely ensures that $E[\chi|\xi]$ is a suitable notion of error between the $\BV$ solution $\chi$ and the strong solution $\bar \chi$: In addition to providing a tilt-excess-like control of the error, it also provides an estimate on the distance of the interfaces.

On the other hand, the calibration structure $\xi_{i,j}=\xi_i-\xi_j$ ensures that the error functional \eqref{ErrorFunctional} may be rewritten as an expression involving only two contributions: First, the total interface energy of the $\BV$ solution $E[\chi]$ and, second, a linear functional of the characteristic functions $\chi_i$ of the phases:
Indeed, we may compute
\begin{align}
\label{RelativeEntropyRewritten}
E[\chi|\xi]
&=
\frac{1}{2}\sum_{i,j=1,i\neq j}^{P} \int_{I_{i,j}} 1 - \xi_{i,j} \cdot \vec{n}_{i,j} \,\mathrm{d}\mathcal{H}^{d-1}
\\&
\nonumber
=E[\chi] - \frac{1}{2} \sum_{i,j=1,i\neq j}^P \int_{I_{i,j}} (\xi_i-\xi_j) \cdot \vec{n}_{i,j} \,\mathrm{d}\mathcal{H}^{d-1}
\\&
\nonumber
=E[\chi] - \sum_{i,j=1,i\neq j}^P \int_{I_{i,j}} \xi_i \cdot \vec{n}_{i,j} \,\mathrm{d}\mathcal{H}^{d-1}
\\&
\nonumber
=E[\chi] + \sum_{i=1}^P \int_{\Rd} \xi_i \cdot \,\mathrm{d}\nabla \chi_i
\\&
\nonumber
=E[\chi] - \sum_{i=1}^P \int_{\Rd} \chi_i \nabla \cdot \xi_i \,\mathrm{d}x.
\end{align}

This enables us to estimate the time evolution of the error functional $E[\chi|\bar \chi]$ using only two ingredients, namely, first, the sharp energy dissipation estimate \eqref{EnergyDissipationInequalityBVSolution} for the interface energy $E[\chi]$ for $\BV$ solutions, and, second, the evolution equation \eqref{EvolutionPhasesBVSolution} for the phase indicator functions $\chi_i$ from the $\BV$ formulation of mean curvature flow. The equations \eqref{FirstEq}--\eqref{ThirdEq} are crucial for deriving a Gronwall-type estimate for $E[\chi|\xi]$ in subsequent rearrangements. We remark that this approach may be regarded as an instance of the relative entropy method introduced independently by Dafermos \cite{Dafermos} and Di~Perna \cite{DiPerna}.

Note that locally at a two-phase interface or at a triple junction of the strong solution, for any fixed time $t$ the blowups of our vector fields $\xi_{i}(\cdot,t)$ turn out to precisely be calibrations of the planar interface or the triple junction, respectively. However, on a global (not blown-up) scale, the vector fields $\xi_i$ may be thought of as deformed variants of classical calibrations which follow the (smooth but typically curved) interface of the strong solution; we refer to Figure~\ref{FigureCurveCalibrations} and Figure~\ref{fig:triple_rotation} for the illustration of a vector field $\xi_{i,j}=\xi_i-\xi_j$ at a two-phase interface and at a triple junction, respectively.

\begin{figure}
\includegraphics[scale=0.55]{energy_landscape}
\caption{An illustration of the energy landscape interpretation of our construction: The gradient flow calibration provides a smooth lower bound for the rough energy landscape of the interface energy functional $E[\chi]$, capturing the energy and its subgradient at the current configuration correctly. \label{EnergyLandscapeInterpretation}}
\end{figure}

Let us finally comment on the energy landscape interpretation of our approach, as illustrated in Figure~\ref{EnergyLandscapeInterpretation}. Let $S(t)$ be a classical solution to the gradient flow of the interface energy functional, i.e., a classical solution to multiphase mean curvature flow. For each point in time, the ``calibration for the gradient flow'' gives a smooth lower bound
\begin{align*}
\mathcal{F}_t:=\frac{1}{2}\sum_{i,j=1,i\neq j}^P \int_{I_{i,j}} \xi_{i,j}\cdot \vec{n}_{i,j} \,\mathrm{d} \mathcal{H}^{d-1} \overset{\eqref{RelativeEntropyRewritten}}{=} \sum_{i=1}^P \int_{\Rd} \chi_i \nabla \cdot \xi_i \,\mathrm{d} x
\end{align*}
(illustrated as the blue wireframe plot in Figure~\ref{EnergyLandscapeInterpretation}) for the rough landscape of the interface energy functional $E[\chi]$ (illustrated as the colored surface plot in Figure~\ref{EnergyLandscapeInterpretation}). This lower bound is sharp for the interface $S(t)$ in the sense $\mathcal{F}_t[S(t)]=E[S(t)]$, and it describes the local direction and speed of steepest descent of the energy functional $E[\chi]$ at $S(t)$ correctly in the sense $D\mathcal{F}_t[S(t)]\in DE[S(t)]$ (where heuristically $DE$ denotes the subdifferential of $E$). Moreover, for each $\chi$ the difference $E[\chi]-\mathcal{F}_t[\chi] =E[\chi| \xi] $ provides an estimate for the error between the smooth solution $S(t)$ and the configuration $\chi$, as measured in a tilt-excess-like quantity.

\section{Main results}

\subsection{Weak-strong uniqueness principle}
In the following, we present our weak-strong uniqueness principle for
$\BV$-solutions of multiphase mean curvature flow in the plane. In addition, we provide a quantitative stabiity estimate, i.\,e., as long as a strong solution exists, any solution to the $\BV$ formulation of multiphase mean curvature
flow with slightly perturbed initial data remains close to it.
Our results are valid under minimal assumptions on the surface tensions, see Definition~\ref{DefinitionAdmissibleSurfaceTensions}.

\begin{theorem}[Weak-strong uniqueness and quantitative stability]\label{MainResult}
Let $d=2$ and $P\in \mathbb{N}$, $P\geq 2$.
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution of multiphase mean curvature flow on $\Rd[d]$ in the sense of Definition~\ref{DefinitionStrongSolution} on some time interval $[0,T_{\mathrm{strong}})$.
Then any $\BV$ solution $\chi=(\chi_1,\ldots,\chi_P)$ to multiphase mean curvature flow on $\Rd$ in the sense of Definition~\ref{DefinitionBVSolution} must coincide with the strong solution $\bar \chi$ for all $0\leq t<T_{\mathrm{strong}}$, provided that it starts from the same initial data.

Furthermore, the evolution by mean curvature is stable with respect to perturbations in the initial data: For a general $\BV$ solution $\chi$ to multiphase mean curvature flow in the sense of Definition~\ref{DefinitionBVSolution}, the stability estimate
\begin{align*}
E[\chi|\xi](T) \leq E[\chi|\xi](0)e^{CT}
\end{align*}
holds true for almost every $T\in[0,T_{\mathrm{strong}})$. Here, $C>0$ only depends on $\bar \chi$. The interface error functional $E[\chi|\xi]$ is defined in \eqref{DefinitionRelativeEntropy}, with $\xi_{i,j}$ denoting the gradient flow calibration for the classical solution $\bar\chi$ as constructed in Proposition~\ref{PropositionExistenceCalibration}.
\end{theorem}

\begin{proof}
	Theorem~\ref{MainResult} is an immediate consequence of Proposition~\ref{PropositionUniqueness}, Proposition~\ref{PropositionExistenceCalibration}, and Proposition~\ref{PropositionVolumeError}.
\end{proof}

\subsection{Calibrations and inclusion principle}
The key ingredient for our uniqueness result prior to topology changes is the following gradient flow analogue of the notion of calibrations for minimal partitions. Our main result, Theorem~\ref{MainResult}, is then an immediate consequence of two implications: First, the existence of a gradient flow calibration guarantees uniqueness of the $\BV$ solution (see Proposition~\ref{PropositionUniqueness} and Proposition~\ref{PropositionVolumeError}); second, the existence of a gradient flow calibration is guaranteed as long as a classical solution to multiphase mean curvature flow exists (see Proposition~\ref{PropositionExistenceCalibration}).

\begin{definition}[Calibrations for the gradient flow]
\label{DefinitionCalibrationGradientFlow}
Let $d\geq 2$, $P\geq 2$ be integers and let $\sigma\in\Rd[P\times P]$ be an admissible
matrix of surface tensions in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions}.
Let $T>0$ and let $\smash{{(\bar \Omega_1(t),\ldots,\bar \Omega_P(t))}_{t\in[0,T]}}$ be such that for all $t\in [0,T]$  we have that $(\bar \Omega_1(t),\ldots,\bar \Omega_P(t))$ is a partition of finite surface energy of $\Rd$ in the sense of Definition~\ref{DefinitionPartition}.
For any $i,j\in \{1,\ldots,P\}$ with $i\neq j$ let ${\bar{I}}_{i,j}:=\partial^* {\bar{\Omega}}_i\cap \partial^* {\bar{\Omega}}_j$ be the interface between the phases $i$ and $j$.
We then call a tuple consisting of vector fields $\xi_i\in C^1_{\mathrm{cpt}}(\Rd\times[0,T];\Rd[d])$, $1\leq i\leq P$, and a vector field $B\in C^1_{\mathrm{cpt}}(\Rd\times[0,T];\Rd[d])$ a \emph{calibration for the gradient flow} for the partition $\smash{(\bar\Omega_1(t),\ldots,\bar\Omega_P(t))_{t\in[0,T]}}$ if the following conditions are satisfied:
\begin{subequations}
\begin{itemize}
\item For each pair of phases $i,j\in\{1,\ldots, P\}$, the vector field
\begin{align}
\label{Calibrations}
\xi_{i,j}:=\frac{1}{\sigma_{ij}}(\xi_i-\xi_j)
\end{align}
coincides on ${\bar{I}}_{i,j}$ with the normal $\bar{\vec{n}}_{ij}$ (with the convention that $\bar{\vec{n}}_{ij}$ points from phase $i$ into phase $j$) and
satisfies an estimate of the form
\begin{align}
\label{LengthControlXi}
|\xi_{i,j}(x,t)|\leq \max\{1-c \dist^2(x,{\bar{I}}_{i,j}(t)),\,0\}
\end{align}
for some $c>0$.
\item The evolution of the vector fields $\xi_{i,j}$ is approximately transported by the velocity field $B$ in the sense
\begin{align}
\label{TransportEquationXi}
\big|\partial_t \xi_{i,j} + (B\cdot \nabla) \xi_{i,j} + (\nabla B)^{\mathsf{T}} \xi_{i,j} \big|(x,t)
\leq C \dist(x,{\bar{I}}_{i,j}(t))
\end{align}
and
\begin{align}
\label{LengthConservation}
\big|\partial_t |\xi_{i,j}|^2 + (B\cdot \nabla) |\xi_{i,j}|^2 \big|(x,t)
\leq C \dist^2(x,{\bar{I}}_{i,j}(t))
\end{align}
for some $C>0$.
\item The normal component of the velocity field $B$ near the interface ${\bar{I}}_{i,j}$ is approximately given by the mean curvature of ${\bar{I}}_{i,j}$ in the sense that
\begin{align}
\label{Dissip}
\big|\xi_{i,j}\cdot B+\nabla \cdot \xi_{i,j}\big|
\leq
C \dist(x,{\bar{I}}_{i,j}(t))
\end{align}
for some $C>0$.
\end{itemize}
\end{subequations}
\end{definition}
Note that, at least heuristically, such a calibrated flow is a solution to mean curvature flow due to \eqref{Dissip} evaluated at $x\in {\bar{I}}_{i,j}$ and the interpretation of $B$ as the velocity field.

The next proposition states that for general $d\geq 2$ the existence of a gradient flow calibration for a given partition of $\Rd$ into $P$ domains $(\bar \Omega_1(t),\ldots,\bar \Omega_P(t))$ constrains the possible locations of the interfaces in weak ($\BV$) solutions to mean curvature flow to the corresponding interfaces of the partition $(\bar \Omega_1,\ldots,\bar \Omega_P)$. This assertion may be seen as a multiphase analogue of the varifold comparison principle by Ilmanen \cite[Theorem 10.7]{Ilmanen}, which provides a corresponding inclusion given a level set solution to two-phase mean curvature flow and any Brakke solution. Note that such an inclusion does not yet yield uniqueness of $\BV$ solutions, as it does not exclude the possibility that suddenly all except for one phase disappear.
\begin{proposition}[Quantitative inclusion principle]
\label{PropositionUniqueness}
Let $d\geq 2$ and $P\geq 2$ be integers and let $\sigma \in \Rd[P\times P]$ be an admissible matrix of surface tensions, see Definition~\ref{DefinitionAdmissibleSurfaceTensions}.
Let $T>0$ and let $\smash{(\bar\Omega_1(t),\ldots,\bar \Omega_P(t))_{t\in[0,T]}}$ be an evolving partition of finite surface energy of $\Rd$, see Definition~\ref{DefinitionPartition}.
Let a calibration for the gradient flow in the sense of Definition~\ref{DefinitionCalibrationGradientFlow} exist for the partition $\smash{(\bar{\Omega}_1(t),\ldots,\bar{\Omega}_P(t))_{t\in[0,T]}}$. Then the interfaces $I_{i,j}(t) := \partial^* \{\chi_i(t)=1\} \cap \partial^* \{\chi_j(t)=1\}$ of any $\BV$ solution to mean curvature flow $(\chi_1,\ldots,\chi_P)$ in the sense of
Definition~\ref{DefinitionBVSolution} with the same initial data as the calibrated partition
must be contained in the corresponding interfaces ${\bar{I}}_{i,j}(t):=\partial^* \bar \Omega_i(t)\cap \partial^* \bar \Omega_j(t)$ for a.\,e.\ $0<t<T$, i.e.\ it holds that $I_{i,j}(t)\subset {\bar{I}}_{i,j}(t)$ for all $i,j$ with $i\neq j$ up to $\mathcal{H}^{d-1}$ null sets.

Furthermore, the existence of a gradient flow calibration also implies a stability estimate: Introducing the interface error functional
\begin{align}
\label{DefinitionRelativeEntropy}
E[\chi|\xi](t):=
\sum_{i,j=1,i\neq j}^P \sigma_{i,j} \int_{I_{i,j}(t)}
1-\xi_{i,j}(\cdot,t) \cdot \vec{n}_{i,j}(\cdot,t) \,\mathrm{d}\mathcal{H}^{d-1},
\end{align}
we have the stability estimate
\begin{align}\label{StabilityEstimate}
E[\chi|\xi](t) \leq E[\chi|\xi](0)e^{Ct}
\end{align}
for almost every $t\in[0,T)$ and for some $C>0$ depending on the calibrated partition.
\end{proposition}

\subsection{Gradient flow calibrations for regular networks}

In fact, in the planar case the existence of a classical solution to mean curvature flow -- in the sense of a smooth evolution of curves meeting at triple junctions with the correct contact angle, see Definition~\ref{DefinitionStrongSolution} -- entails the existence of a calibration for the gradient flow:
\begin{proposition}
\label{PropositionExistenceCalibration}
Let $d=2$ and $P \in \mathbb{N}$, $P\geq 2$. Let $(\bar{\Omega}_1(t),\ldots,\bar{\Omega}_P(t))_{t\in[0,T)}$ be a smooth solution to multiphase mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution}. Then there exists a gradient flow calibration in the sense of Definition~\ref{DefinitionCalibrationGradientFlow}.
\end{proposition}

We furthermore sharpen Proposition~\ref{PropositionUniqueness} in the planar case to also exclude the sudden vanishing of phases.
\begin{proposition}
\label{PropositionVolumeError}
Let $d=2$ and $P \in \mathbb{N}$, $P\geq 2$. Let $(\bar \Omega_1(t),\ldots,\bar \Omega_P(t))_{t\in[0,T)}$ be a strong solution to multiphase mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution}
and let $\bar\chi_i:=\chi_{\bar \Omega_i}$ denote
the characteristic function of the $\mathcal{L}^d$-measurable set $\bar \Omega_i$ for $i = 1,\ldots, P$.
Let $(\chi_1,\ldots,\chi_P)$ be a $\BV$ solution to multiphase mean curvature flow in the sense of Definition~\ref{DefinitionBVSolution}.
Introduce the bulk error functional
\begin{align}
\label{BulkError}
E_{\mathrm{volume}}[\chi|{\bar \Omega}]:=\sum_{i=1}^P \int_{\Rd} |\chi_i-{\bar\chi}_i| \min\big\{\dist\big(x,\partial \bar{\Omega}_i\big),1\big\} \,\mathrm{d}x.
\end{align}
Then the stability estimate
\begin{align}
\label{StabilityEstimateVolume}
E_{\mathrm{volume}}[\chi|{\bar \Omega}]
\leq C e^{CT} \big(E_{\mathrm{volume}}[\chi| \bar \Omega ](0)+E[\chi|\xi](0)\big)
\end{align}
holds, where $E[\chi|\xi]$ is defined in \eqref{DefinitionRelativeEntropy} and $C>0$ only depends on $\bar \Omega$.
\end{proposition}

\subsection{Basic definitions}

In the following, we recall the precise definitions of the solution concepts for multiphase mean curvature flow which our main results are concerned with.
We begin with the notion of admissible surface tensions.

\begin{definition}[Admissible matrix of surface tensions]\label{DefinitionAdmissibleSurfaceTensions}
Let $P\geq 2$ be an integer and $\sigma=(\sigma_{i,j})_{i,j=1,\ldots,P}\in \Rd[P\times P]$ be a matrix. The matrix $\sigma$
is called an \emph{admissible matrix of surface tensions} if the following conditions are satisfied:
\begin{subequations}
\begin{itemize}[leftmargin=0.7cm]
\item[i)] (Symmetry) It holds that $\sigma_{i,j}=\sigma_{j,i}$ and $\sigma_{i,i}= 0$ for every $i,j\in\{1,\ldots,P\}$.
\item[ii)] (Positivity) We have $\sigma_{\mathrm{min}}:=\min\{\sigma_{i,j}\colon i,j\in\{1,\ldots,P\},i\neq j\}>0$.
\item[iii)] (Coercivity) For all choices of pairwise distinct $i,j,k\in\{1.\ldots, P\}$ the following strict
triangle inequality holds true
\begin{align}\label{TriangleInequalitySurfaceTensions}
\sigma_{i,j} < \sigma_{i,k} + \sigma_{k,j}.
\end{align}
Moreover, we assume the symmetric matrix $Q$ with entries
\begin{align*}
 Q_{i,j} := \sigma_{P,i}^2 + \sigma_{P,j}^2 - \sigma_{i,j}^2, \quad i,j=1,\dots,P-1
\end{align*}
to be strictly positive definite:
\begin{align}\label{NoNucleationSurfaceTensions}
	z^{\mathsf{T}}Qz >0 \quad \text{for all } z\in \Rd[P-1]\setminus\{0\}.
\end{align}
\end{itemize}
\end{subequations}
\end{definition}

We labeled \eqref{TriangleInequalitySurfaceTensions} and \eqref{NoNucleationSurfaceTensions} as coercivity properties
for the following reasons. First, the strict triangle inequality \eqref{TriangleInequalitySurfaceTensions} will ensure that
our relative entropy functional provides control on wetting, i.e., the nucleation of a third phase at the smooth part
of an interface between two phases. Second, the assumption \eqref{NoNucleationSurfaceTensions}
turns out to be a condition for our relative entropy functional
to prevent the nucleation of a fourth phase (or clusters of phases) at a triple junction.
This condition also appeared in \cite{Lawlor-Morgan} and
is in fact equivalent to the $l^2$-embeddability of the matrix of surface tensions $\sigma$ into $\Rd[P-1]$, cf.\ \cite{Schoenberg35}, in the following sense: There exist points $q_1$,\ldots, $q_{P} \in \Rd[P-1]$ such that $\sigma_{i,j}=|q_i - q_j|$ for all $i,j\in \{1,\ldots,P\}$, see Figure~\ref{fig:l2embedding}.

\begin{figure}
	\subcaptionbox{\label{fig:herring}}{
		\begin{tikzpicture}[scale=3]
		  	\tikzmath{\l = .6;};
		  	\draw[-Latex] (0,0) -- (0:\l);
		  	\node at ($(0:\l+.1)$) {$\vec{n}_{k,i}$};
		  	\draw[-Latex] (0,0) -- (130:\l);
		  	\node at ($(130:\l+.05)$) {$\vec{n}_{i,j}$};
		  	\draw[-Latex] (0,0) -- (210:\l);
		  	\node at ($(210:\l+.1)$) {$\vec{n}_{j,k}$};
		\end{tikzpicture}
	}
	\subcaptionbox{\label{fig:l2embedding}}{
		\begin{tikzpicture}[scale=5]
			\tikzmath{\l = .6;};
			\coordinate (A) at (0,0);
			\coordinate (B) at (30:\l);
			\draw [name path=A--B,color=white] (A) -- (B);
			\coordinate (C) at (0:\l);
			\coordinate (D) at ($(0:\l)+(130:\l)$);
			\draw [name path=C--D,color=white] (C) -- (D);
			\coordinate (E) at (intersection of A--B and C--D);

		  	\draw[-Latex] (0,0) -- node[label= below:{$\sigma_{k,i}$}] {} (0:\l);
		  	\draw[-Latex] (0:\l) --  node[label= right:{$\sigma_{i,j}$}] {}  (E);
		  	\draw[-Latex] (E) --node[label= left:{$\sigma_{j,k}$}] {}  (0,0);

		  	\filldraw (0,0) circle[radius=.25pt];
			\node[below, outer sep=2pt] (0,0) {$q_k$};
			\filldraw (\l,0) circle[radius=.25pt];
			\node[below, outer sep=2pt] at (\l,0) {$q_i$};
			\filldraw (E) circle[radius=.25pt];
			\node[above, outer sep=2pt] at (E) {$q_j$};

		\end{tikzpicture}
	}
	\caption{a) Normals $\vec{n}_{i,j}$, $\vec{n}_{j,k}$ and $\vec{n}_{k,i}$ satisfying the balance-of-forces condition $ \sigma_{i,j}\vec{n}_{i,j} + \sigma_{j,i}\vec{n}_{j,i} + \sigma_{k,i}\vec{n}_{k,i}=0$. b) Sketch of the points $q_i$, $q_j$ and $q_k$ of the $l^2$-embedding of $\sigma$.}
\end{figure}

\begin{definition}[Partitions with finite interface energy, cf.\ \cite{AmbrosioFuscoPallara}]
\label{DefinitionPartition}
Let
$P\geq 2$ be an integer and let $\sigma\in\Rd[P\times P]$ be an admissible
matrix of surface tensions in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions}. Let $(\Omega_1,\ldots,\Omega_P)$ be a partition of $\Rd$ in the sense that $\Omega_i \subset \Rd$ and $\Rd \setminus \smash{\bigcup_{i=1}^P} \Omega_i$ is a set of $\mathcal{L}^d$-measure zero. Let $\chi_i:=\chi_{\Omega_i}$ denote
the characteristic function of the $\mathcal{L}^d$-measurable set $\Omega_i$ for $i = 1,\ldots, P$.
We call $\chi=(\chi_1,\ldots,\chi_P)$ (or equivalently $(\Omega_1,\dots,\Omega_P)$) a \emph{partition of $\Rd$ with finite interface energy} if the energy
\begin{align}\label{EnergyFunctionalBVPartition}
E[\chi] := \sum_{i,j=1,i\neq j}^P \sigma_{i,j}\int_{\Rd}\frac{1}{2}
\big(\mathrm{d}|\nabla\chi_i|+\mathrm{d}|\nabla\chi_j|-\mathrm{d}|\nabla(\chi_i{+}\chi_j)|\big)
\end{align}
is finite.
\end{definition}
Note that for a partition of $\Rd$ with finite interface energy,
each $\Omega_i$ is a set of finite perimeter. By introducing the
interfaces $I_{i,j}:=\partial^*\Omega_i\cap\partial^*\Omega_j$ as the intersection
of the respective reduced boundaries, the energy of a partition $\chi$ can be
rewritten in the equivalent form
\begin{align}
\label{eq:energy}
E[\chi] =  \sum_{i,j=1,i\neq j}^P \sigma_{i,j}\int_{I_{i,j}}1\,\mathrm{d}\mathcal{H}^{d-1}.
\end{align}
We next recall the notion of $\BV$ solutions to multiphase mean curvature flow, cf.\ \cite{LauxOtto, LauxOttoBrakke}.

\begin{definition}[BV solution for multiphase mean curvature flow]\label{DefinitionBVSolution}
Let $d\geq 2$ and $P\geq 2$ be integers. Let $\sigma\in\Rd[P\times P]$ be an admissible matrix of surface tensions
in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions}, and let $T_{\BV}>0$ be a finite time horizon.
Let $\chi_0=(\chi_{0,1},\ldots,\chi_{0,P})$ be an initial partition of $\Rd$ with finite interface energy in the sense of Definition~\ref{DefinitionPartition}.

A measurable map
\begin{align*}
\chi=(\chi_1,\ldots,\chi_P)\colon \Rd \times [0,T_{\BV})\to\{0,1\}^P
\end{align*}
(or the corresponding tupel of sets $\Omega_i(t):=\{\chi_i(t) =1\}$ for $i=1,\ldots, P$)
is called a \emph{BV solution for multiphase mean curvature flow with initial data $\chi_0$}
if the following conditions are satisfied:
\begin{subequations}
\begin{itemize}[leftmargin=0.7cm]
\item[i)] (Partition with finite interface energy) For almost every $T\in [0,T_{\BV})$, $\chi(T)$
is a partition of $\Rd$ with finite interface energy in the sense of Definition~\ref{DefinitionPartition} and
\begin{align}\label{GlobalEnergyBoundBVSolution}
\esssup_{T\in [0,T_{\BV})} E[\chi(\cdot,T)]
= \esssup_{T\in [0,T_{\BV})} \sum_{i,j=1,i\neq j}^P \sigma_{i,j}\int_{I_{i,j}(T)}1\,\mathrm{d}\mathcal{H}^{d-1} < \infty,
\end{align}
where $I_{i,j}=\partial^* \Omega_i \cap \partial^* \Omega_j$ for $i\neq j$ is the interface between $\Omega_i$ and $\Omega_j$.
\item[ii)] (Evolution equation) For all $i\in \{1,\ldots,P\}$, there exist normal velocities $V_i\in L^2(\Rd\times [0,T_{\BV}),|\nabla\chi_i|\otimes\mathcal{L}^1)$
 in the sense that each $\chi_i$ satisfies the evolution equation
\begin{align}\label{EvolutionPhasesBVSolution}
\nonumber
&\int_{\Rd} \chi_i(\cdot,T) \varphi(\cdot,T) \,\mathrm{d}x
-\int_{\Rd} \chi_{0,i} \varphi(\cdot,0) \,\mathrm{d}x
\\&
=\int_0^T \int_{\Rd} V_i \varphi \,\mathrm{d}|\nabla \chi_i| \,\mathrm{d}t
+ \int_0^T \int_{\Rd} \chi_i \partial_t \varphi \,\mathrm{d}x \,\mathrm{d}t
\end{align}
for almost every $T\in [0,T_{\BV})$ and all $\varphi\in C^\infty_{\mathrm{cpt}}(\Rd \times [0,T_{\BV}])$.
Moreover, the (reflection) symmetry condition
$V_i\cdot\smash{\frac{\nabla \chi_i}{|\nabla \chi_i|}}=V_j\cdot \smash{\frac{\nabla \chi_j}{|\nabla \chi_j|}}$
shall hold $\mathcal{H}^{d-1}$-almost everywhere on
the interfaces $I_{i,j}$ for $i\neq j$.
\item[iii)] (BV formulation of mean curvature) The normal velocities satisfy the equation
\begin{align}\label{BVFormulationMeanCurvature}
\nonumber
&\sum_{i,j=1,i\neq j}^P \sigma_{i,j}\int_0^{T_{\BV}}\int_{I_{i,j}(t)}V_i \frac{\nabla \chi_i}{|\nabla \chi_i|} \cdot \vec{B}
\,\mathrm{d}\mathcal{H}^{d-1} \,\mathrm{d}t
\\&
= -\sum_{i,j=1,i\neq j}^P \sigma_{i,j}\int_0^{T_{\BV}}\int_{I_{i,j}(t)}
\bigg(\operatorname{Id}-\frac{\nabla \chi_i}{|\nabla \chi_i|}\otimes \frac{\nabla \chi_i}{|\nabla \chi_i|}\bigg) : \nabla \vec{B}
\,\mathrm{d}\mathcal{H}^{d-1} \,\mathrm{d}t
\end{align}
for all $B\in C^\infty_{\mathrm{cpt}}(\Rd \times [0,T_{\BV}];\Rd)$.
\item[iv)] (Energy dissipation inequality) The sharp energy dissipation inequality
\begin{align}\label{EnergyDissipationInequalityBVSolution}
E[\chi(\cdot, T)] + \sum_{i,j=1,i\neq j}^P \sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
|V_i|^2 \,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\leq E[\chi_0]
\end{align}
holds true for almost every $T\in [0,T_{\BV})$.
\end{itemize}
\end{subequations}
\end{definition}

Next, we give the definition of strong solutions to multiphase mean curvature flow. To this end, we first define a notion
of regular partitions and regular networks of interfaces (cf.\ \cite{MantegazzaNovagaPludaSchulze}).

\begin{definition}[Regular partitions and networks of interfaces]
\label{DefinitionRegularPartition}
Let $d=2$, let $P\geq 2$ be an integer,
and let $(\bar{\Omega}_1,\ldots,\bar{\Omega}_P)$ be a partition with finite interface energy of open subsets of $\Rd[2]$ such that $\partial^* \bar \Omega_i = \partial \bar \Omega_i$.
Moreover, let $\bar\chi_i:=\chi_{\bar{\Omega}_i}$ denote
the characteristic function of the $\mathcal{L}^d$-measurable set ${\bar{\Omega}}_i$, and
let ${\bar{I}}_{i,j}:=\partial \bar{\Omega}_i\cap\partial \bar{\Omega}_j$ denote the respective interfaces for $i\neq j$.
We call $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ (or equivalently $(\bar \Omega_1,\ldots, \bar \Omega_P)$) a \emph{regular partition of $\Rd[2]$}
and $({\bar{I}}_{i,j})_{i\neq j}$ a \emph{regular network of interfaces in $\Rd[2]$} if the
following properties are satisfied:
\begin{itemize}[leftmargin=0.7cm]
\item[i)] (Regularity) Each interface ${\bar{I}}_{i,j}$ is a one-dimensional manifold
with boundary of class $C^4$. The interior of each interface is embedded.
\item[ii)] (Multi-points are triple junctions) Two different interfaces may intersect only at their boundary. Moreover,
at each intersection point exactly three interfaces meet. In other words, all multi-points
of the network of interfaces $({\bar{I}}_{i,j})_{i\neq j}$ are triple junctions.
\item[iii)] (Balance-of-forces condition) Let $p\in \Rd[2]$ be
a triple junction present in the network. Assume for notational
concreteness that at the triple junction $p$, the three phases $\bar{\Omega}_i$, $\Omega_j$
and $\bar{\Omega}_l$ meet. Then, the balance-of-forces condition.
\begin{align}\label{HerringAngleCondition}
\sigma_{i,j}\bar{\vec{n}}_{i,j}(p) + \sigma_{j,k}\bar{\vec{n}}_{j,k}(p) + \sigma_{k,i}\bar{\vec{n}}_{k,i}(p) = 0
\end{align}
has to be satisfied. Here, $\bar{\vec{n}}_{i,j}(x)$ denotes the unit-normal vector field of the interface
${\bar{I}}_{i,j}(t)$ at $x\in \bar I_{i,j}$ pointing from phase $\Omega_i(t)$ towards phase $\bar{\Omega}_j(t)$.
\end{itemize}
Let $\sigma\in\Rd[P\times P]$ be an admissible
matrix of surface tensions in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions}.
We call $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ (or equivalently $(\bar \Omega_1,\ldots, \bar \Omega_P)$) a \emph{regular partition of $\Rd[2]$ with finite interface energy},
if $\bar\chi$ satisfies in addition to the previous requirements
\begin{align}\label{EnergyFunctionalRegularPartition}
E[\bar\chi] := \sum_{i,j=1,i\neq j}^P \sigma_{i,j}\int_{{\bar{I}}_{i,j}}1\,\mathrm{d}S < \infty.
\end{align}
\end{definition}

\begin{figure}
		\begin{tikzpicture}[scale=.8]

		\draw plot [smooth] coordinates { (0:3)(9.5:2)(50:1.4)};
		\draw plot [smooth] coordinates { (50:1.4)(60:2)(86.15:3)(90:3.5)};
		\draw plot [smooth] coordinates { (50:1.4)(70:1) (130:0.5)};
		\draw plot [smooth] coordinates { (280:.5) (200:.2) (130:0.5)};
		\draw plot [smooth] coordinates {(130:0.5)(137:1.5)(140:2.5)};

		\draw plot [smooth] coordinates {(140:2.5)(120:3.5)(105:3.8)(90:3.5)};
		\draw plot [smooth] coordinates {(0:3)(7:3.5)(20:3.8)(45:4)(75:4)(85:3.82)(90:3.5)};
		\draw plot [smooth] coordinates {(140:2.5)(150:2.8)(158:3)(175:3)(203.5:2.7)(220:2.4)(250:1.3)};
	  \draw plot [smooth] coordinates {(250:1.3)(250:1.1)(255:0.8)(280:.5)};
		\draw plot [smooth] coordinates {(250:1.3)(275:1.8)(310:3.3)(333:3.4)(340:3)};
		\draw plot [smooth] coordinates {(280:.5)(320:1)(330:2.3)(340:3)};
	  \draw plot [smooth] coordinates {(340:3)(350:3.05)(0:3)};

		\node at (350:2){${\bar{\Omega}}_i$};

		\node at (100:1.75){${\bar{\Omega}}_j$};

		\node at (45:.8){${\bar{I}}_{i,j}$};

	\end{tikzpicture}
	\caption{Sketch of a regular partition of the plane and the corresponding regular network. \label{fig:partition} }
\end{figure}

The next definition roughly speaking encodes the fact that starting from a regular partition
with an associated regular network of interfaces in the precise sense of the previous definition,
topological changes are excluded on the level of a strong solution.

\begin{definition}[Smoothly evolving partitions and networks of interfaces]
\label{DefinitionSmoothlyEvolvingPartition}
Let $P\geq 2$ be an integer and let
$\bar\chi(0)=(\bar\chi_1(0),\ldots,\bar\chi_P(0))$ be a regular partition of $\Rd[2]$ with
a regular network of interfaces $({\bar{I}}_{i,j}(0))_{i\neq j}$ in the sense of
Definition~\ref{DefinitionRegularPartition}. Let $T_{\mathrm{strong}} > 0$ be a finite time horizon
and consider a family $\big(\bar{\Omega}_1(t),\ldots,\bar{\Omega}_P(t)\big)_{t\in[0,T_{\mathrm{strong}})}$
of regular partitions of $\Rd[2]$ in the sense of Definition~\ref{DefinitionRegularPartition}.
Let $\bar\chi_i(t):=\chi_{\bar{\Omega}_i(t)}$
and ${\bar{I}}_{i,j}(t):=\partial \bar{\Omega}_i(t)\cap\partial \bar{\Omega}_j(t)$ for
all $t\in [0,T_{\mathrm{strong}})$ and each pair $i\neq j$ with $i,j =1,\ldots,P$.

We say that $\bar\chi$ (or equivalently $(\bar \Omega_1,\dots, \bar \Omega_P)$) is a \emph{smoothly evolving regular partition of $\Rd[2]$}
and $({\bar{I}}_{i,j})_{i\neq j}$ is a \emph{smoothly evolving regular network of interfaces in $\Rd[2]$}
if there exists a map
\begin{align*}
\Psi\colon \Rd[2]\times [0,T_{\mathrm{strong}})\to \Rd[2],\,(x,t)\mapsto\Psi(x,t)=\Psi^t(x)
\end{align*}
such that $\bar\chi_i(t) = \bar\chi_i(0)\circ\Psi^t$, ${\bar{I}}_{i,j}(t)=\Psi^t({\bar{I}}_{i,j}(0))$ for all $t\in [0,T_{\mathrm{strong}})$,
$\Psi^0=\mathrm{Id}$, and which is in addition subject to the following regularity conditions:
\begin{itemize}[leftmargin=0.7cm]
\item[i)] For all $t\in [0,T_{\mathrm{strong}})$, the map $\Psi^t\colon \Rd[2]\to \Rd[2]$ is a homeomorphism.
\item[ii)] For all $t\in [0,T_{\mathrm{strong}})$ and all $i\neq j$, the restriction
$\Psi^t_{i,j}\colon {\bar{I}}_{i,j}(0)\to {\bar{I}}_{i,j}(t)$ is a $C^4$-diffeomorphism such that
$\|\Psi^t_{i,j}\|_{L^\infty_tW^{4,\infty}_x}<\infty$.
\item[iii)] For all $i\neq j$, the restriction $\Psi_{i,j}\colon {\bar{I}}_{i,j}(0)\times [0,T_{\mathrm{strong}})\to \Rd[2]$
satisfies $\partial_t\Psi_{i,j}\in C^0([0,T_{\mathrm{strong}});C^3({\bar{I}}_{i,j}(0);\Rd[2]))$
as well as $\|\partial_t\Psi_{i,j}\|_{L^\infty_tW^{3,\infty}_x}<\infty$.
\item[iv)] Moreover, we assume that there exists $r_c\in (0,\frac{1}{2}]$ with the following property:
For all $t\in [0,T_{\mathrm{strong}})$, all $i\neq j$ and all points
$x\in {\bar{I}}_{i,j}(t)\cup \partial {\bar{I}}_{i,j}(t)$
there exists a function $g_{i,j}\colon (-1,1)\to\Rd[]$ with $g'_{i,j}(0)=0$ and $y_{min} \in [-1,0]$,
such that after a translation and a rotation, ${\bar{I}}_{i,j}(t)\cap B_{2r_c}(x)$ is given by
the graph $\{(y,g_{i,j}(y))\colon y\in (y_{min},1)\}\cap B_{2r_c}(0)$, which we
additionally ask to be a connected set. Furthermore, for any of these functions $g_{i,j}$
we ask the pointwise bounds $|g_{i,j}^{(m)}|\leq r_c^{-(m-1)}$ to hold for all $1\leq m\leq 3$.
\end{itemize}
\end{definition}

We have everything in place to proceed with the definition of strong solutions for multiphase
mean curvature flow.

\begin{definition}[Strong solution for multiphase mean curvature flow]
\label{DefinitionStrongSolution}
Let $d=2$, let $P\geq 2$ be an integer, $\sigma\in\Rd[P\times P]$ be an admissible matrix of surface tensions
in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions}, $T_{\mathrm{strong}}>0$ be a finite time horizon,
and let $\bar\chi_0=(\bar\chi_1^0,\ldots,\bar\chi_P^0)$ be an initial regular partition of $\Rd[2]$ with finite interface energy
in the sense of Definition~\ref{DefinitionRegularPartition}.

A measurable map
\begin{align*}
\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)\colon \Rd\times [0,T_{\mathrm{strong}})\to\{0,1\}^P
\end{align*}
(or equivalently the tupel of time-dependent sets $\bar \Omega_i(t) := \{\bar \chi_i(\cdot, t) = 1\}$ for $i=1,\ldots,P$ and $t \in [0,T_{\mathrm{strong}})$)
is called a \emph{strong solution for multiphase mean curvature flow with initial data $\bar\chi_0$}
if the following conditions are satisfied:
\begin{subequations}
\begin{itemize}[leftmargin=0.7cm]
\item[i)] (Smoothly evolving regular partition with finite interface energy)
The family $\bar\chi$ is a smoothly evolving regular partition of $\Rd[2]$
and the family $({\bar{I}}_{i,j})_{i\neq j}$ is a smoothly evolving regular network of interfaces in
$\Rd[2]$ in the sense of Definition~\ref{DefinitionSmoothlyEvolvingPartition}. In particular,
for every fixed ${t}\in [0,T_{\mathrm{strong}})$, $\bar\chi({t})$ is a regular partition of $\Rd[2]$
and $(\bar{I}_{i,j}(t))_{i\neq j}$ is a regular network of interfaces in $\Rd[2]$
in the sense of Definition~\ref{DefinitionRegularPartition} such that
\begin{align}\label{GlobalEnergyBoundStrongSolution}
\sup_{t\in [0,T_{\mathrm{strong}})} E[\bar\chi({t})]
= \sup_{t\in [0,T_{\mathrm{strong}})} \sum_{i,j=1,i\neq j}^P \sigma_{i,j}\int_{{\bar{I}}_{i,j}(T)}1\,\mathrm{d}S < \infty.
\end{align}
\item[ii)] (Evolution by mean curvature) For $i,j =1,\ldots, P$ with $i\neq j$, $x\in {\bar{I}}_{i,j}$ and $t\in [0,T_{\mathrm{strong}})$ let $\bar V_{i,j}(x,t)$ denote
the normal speed of a point $x\in {\bar{I}}_{i,j}(t)$, i.e., $\bar V_{i,j}(x):=\bar{\vec{n}}_{i,j}(x,t)\cdot \partial_t\Psi_{i,j}(y,t)$
at $y=(\Psi^t_{i,j})^{-1}(x)\in {\bar{I}}_{i,j}(0)$, where $\Psi_{i,j}$ and $\Psi^t_{i,j}$ are the maps from the definition of a
smoothly evolving regular network of interfaces, see Definition~\ref{DefinitionSmoothlyEvolvingPartition}.
Denoting by $\bar{\vec{H}}_{i,j}(t)$ the mean curvature vector of ${\bar{I}}_{i,j}(t)$, we then assume that
\begin{align}\label{MotionByMeanCurvature}
\bar V_{i,j}(x,t)\bar{\vec{n}}_{i,j}(x,t) = {\bar{\vec{H}}}_{i,j}(x,t),
\quad\text{for all } t\in [0,T_{\mathrm{strong}}),\, x\in {\bar{I}}_{i,j}(t).
\end{align}
	\item[iii)] (Initial conditions) We have $\bar \chi_i(x,0) = \bar \chi_{0,i}(x)$ for all $x\in \Rd$ and each $i=1,\ldots, P$.
\end{itemize}
\end{subequations}
\end{definition}

\subsection{Relative entropy inequality}
The key ingredient for the proof of Proposition~\ref{PropositionUniqueness} is the derivation of a Gronwall-type inequality for the tilt-excess-like error functional \eqref{DefinitionRelativeEntropy}: We aim to derive an estimate of the form
\begin{align}\label{eq:Gronwall}
E[\chi|\xi](T) \leq E[\chi|\xi](0) +C(\xi)\int_0^TE[\chi|\xi](t)\,\mathrm{d}t
\end{align}
for almost all admissible times $T \geq 0$
from which one may infer the desired stability estimate \eqref{StabilityEstimate} by an application of Gronwall's lemma; the weak-strong uniqueness principle then follows by means of the coercivity properties of the relative entropy error functional \eqref{DefinitionRelativeEntropy} and a subsequent estimate for $E_{\mathrm{volume}}[\chi|\bar \chi]$, see Proposition~\ref{PropositionVolumeError}. The following result contains the first key step in the derivation of the Gronwall-type inequality \eqref{eq:Gronwall}; it is valid for general vector fields $\xi_i$ and $B$ with sufficient smoothness (not just for gradient flow calibrations).

\begin{proposition}[Relative entropy inequality]
\label{PropositionRelativeEntropyInequality}
Let $d\geq 2$, $P\geq 2$ be integers, and
let $\sigma\in\Rd[P\times P]$ be an admissible matrix of surface tensions
in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions}.

Let $\chi=(\chi_1,\ldots,\chi_P)$ be a BV solution of multiphase mean curvature flow in
the sense of Definition~\ref{DefinitionBVSolution} on some time interval $[0,T_{\BV})$.
We denote by
\begin{align}\label{UnitNormalsBVSolution}
\vec{n}_{i,j}&:=\frac{\nabla \chi_{j}}{|\nabla \chi_{j}|}
=-\frac{\nabla \chi_i}{|\nabla \chi_i|},
\quad
\mathcal{H}^{d-1}\text{-a.e.\ on }I_{i,j},
\end{align}
the (measure-theoretic) unit normal vector of the interface $I_{i,j}$ pointing
from the $i$-th to the $j$-th phase of the $\BV$ solution. Moreover, let
\begin{align}\label{NormalVelocitiesBVSolution}
V_{i,j} := V_{i} = -V_{j}, \quad
\mathcal{H}^{d-1}\text{-a.e.\ on }I_{i,j}.
\end{align}

Let $(\xi_{i,j})_{i\neq j\in\{1,\ldots,P\}}$ and $(\xi_i)_{i=1,\ldots,P}$
be families of compactly supported vector fields such that
\begin{align*}
\xi_{i,j},\xi_i \in L^\infty([0,T_{\mathrm{strong}});W^{2,\infty}(\Rd;\Rd[d]))\cap
W^{1,\infty}([0,T_{\mathrm{strong}});C^0(\Rd;\Rd[d]))
\end{align*}
as well as $\sigma_{i,j}\xi_{i,j}=\xi_i-\xi_j$ for all $i\neq j$.
Let
\begin{align*}
B\in L^\infty([0,T_{\mathrm{strong}});W^{1,\infty}(\Rd;\Rd[d]))
\end{align*}
be an arbitrary compactly supported vector field.

Consistently with \eqref{DefinitionRelativeEntropy}, define the interface error control
\begin{align}
\label{DefinitionRelativeEntropyFunctional}
E[\chi|\xi](t)
:=
\sum_{i,j=1,i\neq j}^P \sigma_{i,j} \int_{I_{i,j}(t)}
1-\xi_{i,j}(\cdot,t) \cdot \vec{n}_{i,j}(\cdot,t) \,\mathrm{d}\mathcal{H}^{d-1}.
\end{align}
Then the interface error control
is subject to the estimate
\begin{align}
\nonumber
&E[\chi|\xi](T)
\\&\nonumber
+\sum_{i,j=1,i\neq j}^{P} \frac{\sigma_{i,j}}{2}
\int_0^T\int_{I_{i,j}(t)}
|V_{i,j}{+}\nabla\cdot\xi_{i,j}|^2
+|V_{i,j}\vec{n}_{i,j}{-}(B\cdot\xi_{i,j})\xi_{i,j}|^2
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\label{RelativeEntropyInequality}
\leq E[\chi|\xi](0) + R_{\mathrm{dt}} + R_{\mathrm{dissip}}
\end{align}
for almost every $T\in (0,T_{\mathrm{strong}})$. Here, we made use of the abbreviations
\begin{align*}
R_{\mathrm{dt}}
&:= -\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}\big(\partial_t|\xi_{i,j}|^2{+}(B\cdot\nabla)|\xi_{i,j}|^2\big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t \notag
\\&~~~~
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
\big(\partial_t\xi_{i,j}{+}(B\cdot\nabla)\xi_{i,j}{+}(\nabla B)^\mathsf{T}\xi_{i,j}\big)
\cdot(\vec{n}_{i,j}{-}\xi_{i,j})\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t,\\
R_{\mathrm{dissip}} &:=
\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}|(\nabla\cdot\xi_{i,j})+B\cdot\xi_{i,j}|^2
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}|B\cdot\xi_{i,j}|^2(1-|\xi_{i,j}|^2)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
(1-\vec{n}_{i,j}\cdot\xi_{i,j})(\nabla\cdot\xi_{i,j})(B\cdot\xi_{i,j})
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
+\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
\Big((\mathrm{Id}{-}\xi_{i,j}\otimes\xi_{i,j})B\Big)\cdot(V_{i,j}{+}\nabla\cdot\xi_{i,j})\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
+\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
(1-\vec{n}_{i,j}\cdot\xi_{i,j})(\nabla\cdot B)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
(\vec{n}_{i,j}{-}\xi_{i,j})\otimes(\vec{n}_{i,j}{-}\xi_{i,j}):\nabla B
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t.
\end{align*}
\end{proposition}
\subsection{Structure of the paper}
The remaining part of the paper is organized as follows.
Section~\ref{SectionOutlineStrategy} illustrates our strategy at the two most important examples, a smooth interface, and a triple junction.

The subsequent sections are then divided into two parts. In Section~\ref{SectionWeakStrongUniqueness}, we prove the stability of calibrated flows and exploit all properties of our gradient flow calibrations and the \emph{weak solution}: In Subsection~\ref{SectionDerivationRelativeEntropyInequality} we derive the relative entropy inequality in its full generality of Proposition~\ref{PropositionRelativeEntropyInequality}; in Subsection~\ref{SectionProofMainResult}, we prove the quantitative inclusion principle, Proposition~\ref{PropositionUniqueness}; and Subsection~\ref{SectionVolumeError} provides the ingredients for the control of the volume error in Proposition~\ref{PropositionVolumeError}.

The remaining three sections of the manuscript are devoted to the construction of our gradient flow calibrations given a \emph{strong solution}. First, we provide explicit constructions at a smooth interface (Section~\ref{SectionLocalConstructionsTwoPhase}) and at a triple junction (Section~\ref{SectionLocalConstructionsTriod}). These cases do not only serve as model examples but they also form the building blocks for our general construction in Section~\ref{sec:networkConstruction}, where we glue together these local constructions, to obtain a gradient flow calibration for regular networks, which establishes Proposition~\ref{PropositionExistenceCalibration}.

\section{Outline of the strategy}
\label{SectionOutlineStrategy}

	\subsection{Idea of proof for a smooth interface}
	Let us give a brief idea of the proof, ignoring technical difficulties in the
	simplest case of two phases sharing one single interface with $\sigma=1$.
	In that case, it is sufficient to describe the weak solution and the calibrated flow by a single phase $\Omega(t) \subset \Rd$, resp.\ $\bar \Omega(t) \subset \Rd$ for $t\in [0,T_{\mathrm{strong}})$, the first being a set of finite perimeter and the second being a simply connected, smooth set.
	The relative entropy is then simply given by
	\begin{align*}
		E[\chi | \xi ](t)
		= \int_{\partial^* \Omega(t)} (1-\vec{n}\cdot \xi) \,\mathrm{d}\mathcal{H}^{d-1},
	\end{align*}
	which has the interpretation of an oriented excess of the weak solution with respect to the strong one.
	Here $\chi=\chi(x,t)$ denotes the characteristic function of
	$\Omega=\Omega(t)$ and $\vec{n} = -\frac{\mathrm{d} \nabla \chi}{\mathrm{d} |\nabla \chi |}$
	denotes the (measure theoretic) exterior unit normal of $\partial^\ast \Omega(t)$.
	Furthermore, the vector field $\xi(\cdot,t)$ is an extension of the exterior unit normal
	$ \vec{\bar n}(\cdot,t)$ of the calibrated, smooth interface $\bar{I}(t) := \partial \bar \Omega$
	necessitated by the fact that we evaluate it on the weak solution.

	In order to relate the extension $\xi$ to the evolution, we require it to be
	transported along an extension $B$ of the velocity field of $\bar{I}$ in the sense that
	\begin{equation}\label{eq:xitransport2phase}
		\partial_t \xi =-\left( B\cdot \nabla \right)\xi -(\nabla B)^\mathsf{T} \xi + O\big(\dist(x,\bar{I}(t))\big),
	\end{equation}
	which will help make the second term of $R_{\mathrm{dt}}$ small (see Proposition~\ref{PropositionRelativeEntropyInequality} for the definition).
	The extension for $B$ will be done such that it is constant in the ``normal'' $\xi$-direction,
	meaning we have $(\xi \cdot \nabla ) B=0$, and such that the motion law
	$\vec{\bar n} \cdot B = \bar V= \bar H = - \nabla^{\tan} \cdot \vec{\bar n}$ is still approximately
	true away from the interface in the sense that
	\begin{equation}\label{eq:Banddivxi2phase}
		\xi \cdot B = -\nabla \cdot \xi + O\big(\dist(x,\bar{I}(t))\big),
	\end{equation}
	helping with the first term of $R_{\mathrm{dissip}}$.

	As we also want the functional $E[\chi | \xi]$ to ensure that
	$\chi$ cannot be too far away from $\bar \chi$, we allow for $\xi$ to
	be short, i.e., we have $|\xi|\leq 1$, and we ask this effect to be
	transported by $B$ up to quadratic error
	\begin{equation}\label{eq:modxiquadratic}
		\partial_t |\xi|^2 + (B\cdot \nabla )	|\xi|^2 = O\big(\dist^2(x,{\bar{I}}(t))\big),
	\end{equation}
	keeping the first term of $R_{\mathrm{dt}}$ small.

	In the present case of a single interface, the construction of these vector fields
	is straightforward using the signed distance function $s = s(x,t)$
	to the smooth interface $\bar{I}$:
	We set
	\begin{align*}
		\xi := \zeta(s) \nabla s \quad \text{and} \quad  B := - (\Delta s )\xi,
	\end{align*}
	where $\zeta$ is a suitable cut-off function such that $\zeta(s) = 1-s^2$ close to $s=0$.
	Note that since $|\nabla s | =1$, this implies
	\begin{align}\label{EntropyControlsDistance}
		s^2 = 1-\zeta(s) \leq 1-\zeta(s)\,  \vec{n} \cdot \nabla s = 1- \vec{n} \cdot \xi
	\end{align}
	for $s $ small, so that the relative entropy controls the (truncated) $L^2$ distance of the
	weak solution and the calibrated flow.

	In the following heuristic derivation of the relative entropy inequality (from Proposition~\ref{PropositionRelativeEntropyInequality}) in the case of a single interface, we will use the abbreviation $\int_{\partial^\ast \Omega} \cdot := \int_{\partial^\ast \Omega(t)} \cdot \,\mathrm{d}\mathcal{H}^{d-1}$ for the integral along a time slice $\partial^* \Omega$ of the weak solution.
	Recall that $V$ denotes the
	normal velocity of the weak solution characterized by the distributional equation
	$\partial_t \chi = V\left|\nabla \chi\right|$, see \eqref{EvolutionPhasesBVSolution}, so that the sign convention is $V>0$
	for expanding $\Omega$.

\begin{figure}
\begin{tikzpicture}[scale=5]
\draw[color=white] (0,0.45)--(0,-0.22);
		\begin{scope}

			\draw[thick,opacity=1,color=black] (0,0) arc (120:75:1);
		\end{scope}
			\foreach \i in {0,...,9}
			{
				\foreach \j in {-2,...,2}
				{
					\begin{scope}[shift={($(300:1)+(120 - 5*\i :1)+(120-5*\i : .13*\j)$) }]
						\draw[-{Latex[length=1mm]},opacity=1,color=black]  (0,0) -- ($(120-5*\i:.1-\j*\j*0.02+\j*\j*\j*\j*0.0015)$);
					\end{scope}
				}
			}
\end{tikzpicture}
\caption{Illustration of the vector field $\xi$ at a smooth interface ${\bar{I}}(t)$. The vector field $\xi$ extends the unit normal vector field of ${\bar{I}}(t)$ by projection onto ${\bar{I}}(t)$ and multiplication with a cutoff function.\label{FigureCurveCalibrations}}
\end{figure}

	The optimal energy dissipation rate \eqref{EnergyDissipationInequalityBVSolution} and the definition \eqref{EvolutionPhasesBVSolution} of $V$ imply
	\begin{align*}
		\frac{\mathrm{d}}{\mathrm{d}t} E[\chi | \xi]
		= \frac{\mathrm{d}}{\mathrm{d}t} |\partial^*\Omega| -  \frac{\mathrm{d}}{\mathrm{d}t} \int_\Omega (\nabla \cdot \xi ) \,\mathrm{d}x
		\leq -\int_{\partial^\ast \Omega} V^2  - \int_{\partial^\ast \Omega} V\left(\nabla \cdot \xi \right) - \int_{\partial^\ast \Omega} \partial_t \xi \cdot \vec{n}.
	\end{align*}
	Testing the distributional mean curvature flow equation \eqref{BVFormulationMeanCurvature} with the extended velocity field $B$ gives
	\begin{align*}
		0=\int_{\partial^\ast \Omega} V\left(\vec{n}\cdot B \right) + \int_{\partial^\ast \Omega} \left(\operatorname{Id} - \vec{n}\otimes \vec{n}\right) \colon \nabla B
	\end{align*}
	Adding these terms to the right-hand side of the previous inequality yields
	\begin{align*}
		\frac{\mathrm{d}}{\mathrm{d}t} E[\chi | \xi]
		& \leq -\int_{\partial^\ast \Omega}
		\left(
			V^2 +V \left(\nabla \cdot \xi\right) -V\left(\vec{n}\cdot B\right)
		\right)
		+ \int_{\partial^\ast \Omega} (\nabla \cdot B)
		-\int_{\partial^\ast \Omega} \vec{n}\otimes \vec{n}  \colon \nabla  B\\
		& \quad - \int_{\partial^\ast \Omega} \partial_t \xi \cdot \vec{n}.
	\end{align*}
	Note that $B=\left(\xi\cdot B\right)\xi + \left(\operatorname{Id} - \xi \otimes \xi \right) B$, which we interpret as a decomposition of $B$ into ``normal'' and ``tangential'' parts.
	Furthermore, as a result of $(\xi \cdot \nabla)B=0$ we have
	$  -\left(\vec{n}-\xi\right) \otimes  \left(\vec{n}-\xi\right) \colon  \nabla  B
	= - \left(\vec{n}-\xi\right) \cdot   \left(\vec{n} \cdot  \nabla\right)  B
	= - \vec{n}\otimes \vec{n} \colon \nabla B + \vec{n}\cdot (\nabla B)^\mathsf{T} \xi $.
	 We then insert the first identity, complete the squares, and add and subtract $\left( B\cdot \nabla \right)\xi + (\nabla B)^\mathsf{T} \xi$ to make
	$\partial_t \xi +\left( B\cdot \nabla \right)\xi + (\nabla B)^\mathsf{T} \xi$ appear. Together with the second identity we obtain
	\begin{align}
	\label{eq:firstcompletionsquare}
	\notag
		\frac{\mathrm{d}}{\mathrm{d}t} E[\chi | \xi]
		\leq& -\frac12 \int_{\partial^\ast \Omega}
		\left(
			\left(
				V+\nabla \cdot \xi
			\right)^2
			+
			\left|
				V\vec{n} - \left(\xi\cdot B\right)\xi
			\right|^2
		\right)
		\\\notag
		&+\frac12 \int_{\partial^\ast \Omega}
		\left(
			\left(
				\nabla \cdot \xi
			\right)^2
			+
			|\xi|^2\left(
				\xi \cdot B
			\right)^2
		\right)
		+\int_{\partial^\ast \Omega} V\vec{n} \cdot  \left(\operatorname{Id} - \xi \otimes \xi \right) B
		\\\notag
		&+\int_{\partial^\ast \Omega} \left(\nabla \cdot B\right)
		-\int_{\partial^\ast \Omega}   \left(\vec{n}-\xi\right) \otimes  \left(\vec{n}-\xi\right) \colon  \nabla  B
		+\int_{\partial^\ast \Omega} \vec{n} \cdot \left( B \cdot \nabla \right) \xi
		\\
		&- \int_{\partial^\ast \Omega} \left(\partial_t \xi +\left( B\cdot \nabla \right)\xi + (\nabla B)^\mathsf{T} \xi\right)\cdot \vec{n},
	\end{align}
	where the second line collects precisely the terms left after completing the squares.

	By symmetry considerations, we have
	\begin{align*}
		0
		&=  \int_\Omega \nabla \cdot
		\left[
			\nabla \cdot
			\left(
				B\otimes \xi -\xi \otimes B
			\right)
		\right]
		\,\mathrm{d}x
		= \int_{\partial^\ast \Omega} \left[ \nabla \cdot
		\left(
			B\otimes \xi -\xi \otimes B
		\right) \right] \cdot \vec{n} \\
		& = \int_{\partial^\ast \Omega}
		\left[
			\left( \nabla \cdot \xi	\right) \vec{n} \cdot B
			- \left(\nabla \cdot B\right) \vec{n} \cdot \xi
			-\vec{n} \cdot \left( B\cdot \nabla \right)\xi
		\right],
	\end{align*}
	where for the second line we used $(\xi \cdot \nabla)B=0$.
	Now we use $|\xi | \leq 1$ to drop the prefactor $|\xi|^2$ of $(\xi\cdot B)^2$
	in the second right-hand side integral in inequality \eqref{eq:firstcompletionsquare},
	complete the square, and add the above identity to obtain
	\begin{align*}
		\frac{\mathrm{d}}{\mathrm{d}t} E[\chi | \xi ]
		& \leq -\frac12 \int_{\partial^\ast \Omega}
		\left(
		\left(
		V+\nabla \cdot \xi
		\right)^2
		+
		\left|
		V\vec{n} - \left(\xi\cdot B\right)\xi
		\right|^2
		\right)\\
		&\quad +\frac12 \int_{\partial^\ast \Omega}
		\left(
		\nabla \cdot \xi
		+
		\xi \cdot B
		\right)^2
		+ \int_{\partial^\ast \Omega} \left(\nabla \cdot \xi \right) \left(\vec{n}-\xi\right) \cdot B\\
		& \quad +\int_{\partial^\ast \Omega} V\vec{n} \cdot  \left(\operatorname{Id} - \xi \otimes \xi \right) B
		 +\int_{\partial^\ast \Omega} (1-\vec{n}\cdot \xi ) \left(\nabla\cdot  B\right) \\
		& \quad -\int_{\partial^\ast \Omega}   \left(\vec{n}-\xi\right) \otimes  \left(\vec{n}-\xi\right) \colon  \nabla  B - \int_{\partial^\ast \Omega} \left(\partial_t \xi +\left( B\cdot \nabla \right)\xi + (\nabla B)^\mathsf{T} \xi\right)\cdot \vec{n}.
	\end{align*}
	Once more, we decompose $B$ into ``tangential'' and ``normal'' components, and finally arrive at the entropy dissipation inequality
	\begin{align*}
		\frac{\mathrm{d}}{\mathrm{d}t} E[\chi | \xi ]
		\leq& -\frac12 \int_{\partial^\ast \Omega}
		\left(
		\left(
		V+\nabla \cdot \xi
		\right)^2
		+
		\left|
		V\vec{n} - \left(\xi\cdot B\right)\xi
		\right|^2
		\right)\\
		&+\frac12 \int_{\partial^\ast \Omega}
		\left(
		\nabla \cdot \xi
		+
		\xi \cdot B
		\right)^2
		+ \int_{\partial^\ast \Omega} \left(\nabla \cdot \xi \right) \left(\vec{n}\cdot \xi -1\right) \left(\xi \cdot B\right)\\
		&+\int_{\partial^\ast \Omega} \left(\nabla \cdot \xi + V \right) \vec{n} \cdot  \left(\operatorname{Id} - \xi \otimes \xi \right) B \\
		&+\int_{\partial^\ast \Omega} (1-\vec{n}\cdot \xi ) \left(\nabla\cdot  B\right)
		-\int_{\partial^\ast \Omega}   \left(\vec{n}-\xi\right) \otimes  \left(\vec{n}-\xi\right) \colon  \nabla  B\\
		&- \int_{\partial^\ast \Omega} \left(\partial_t \xi +\left( B\cdot \nabla \right)\xi + (\nabla B)^\mathsf{T} \xi\right)\cdot (\vec{n}-\xi)\\
		&+ \int_{\partial^\ast \Omega} \left(\partial_t \xi +\left( B\cdot \nabla \right)\xi \right)\cdot \xi.
	\end{align*}

	 Now let us briefly argue term-by-term that the right-hand side can be controlled by the
	relative entropy $E[\chi|\bar\chi]$, which together with a Gronwall argument and Proposition~\ref{PropositionVolumeError} would yield
	Theorem~\ref{MainResult} for $d=2$. Thanks to \eqref{eq:Banddivxi2phase},
	the first term of the second line is quadratic in $\dist(x,{\bar{I}})$ and therefore
	controlled by the relative entropy due to \eqref{EntropyControlsDistance}. The second integral of the second line is controlled by the relative entropy
	since $\nabla \cdot \xi $ and $\xi\cdot B$ are uniformly bounded. To handle the third line, we use Cauchy-Schwarz and Young, and absorb $\int(\nabla \cdot \xi + V)^2$ in the first integral. The remaining
	integral of $ | (\operatorname{Id} - \xi \otimes \xi) \vec{n}|^2 = |\vec{n} - (\xi \cdot \vec{n}) \xi|^2
	\lesssim |\vec{n}-\xi|^2 + (1-\vec{n}\cdot \xi)^2$ is controlled by the relative entropy.
	Clearly, both terms in the fourth line are controlled by the relative entropy.
	 Finally, the integrals
	in the fifth and sixth lines are quadratic due to \eqref{eq:xitransport2phase} and the
	factor $\vec{n}-\xi$, and \eqref{eq:modxiquadratic}, respectively.

\subsection{Idea of proof for a triple junction}

The second model case is given by a triple junction, say, with equal surface tensions. To illustrate the additional difficulties, we also present the idea of our proof in this case. However, we restrict ourselves to the case $d=2$.

We denote the phases of the \emph{weak} solution by
$\Omega_1$, $\Omega_2$, and $\Omega_3$ with characteristic functions $\chi_1,$ $\chi_2$, and $\chi_3$.
To simplify notation, we identify indices if they are equivalent mod $3$, i.\,e., we define
$\chi_{4}:=\chi_1$, $\chi_5:=\chi_2$, $\chi_0:=\chi_3$, and so on.
Following the notation of Proposition \ref{PropositionRelativeEntropyInequality}, we denote the normal
vector of the interface $I_{i,i+1}=\partial^* \Omega_i\cap \partial^* \Omega_{i+1}$ between phases $i$ and $i+1$ for $i=1,2,3$ in the weak solution by
\begin{align*}
\vec{n}_{i,i+1}&:=\frac{\mathrm{d}\nabla \chi_{i+1}}{\mathrm{d}|\nabla \chi_{i+1}|}
=-\frac{\mathrm{d}\nabla \chi_i}{\mathrm{d}|\nabla \chi_i|}\quad\quad\mathcal{H}^{1}\text{-a.\,e.\ on }\partial^* \Omega_i \cap \partial^* \Omega_{i+1}.
\end{align*}
Its normal velocity is denoted by $V_i$, which is characterized by the distributional
identity $\partial_t \chi_i = V_i |\nabla \chi_i|$.
Additionally, we will consider its restriction $V_{i,i+1}:= V_i|_{I_{i,i+1}}$ to the interface $I_{i,i+1}$ together with the symmetry condition $V_{i+1,i} := - V_{i,i+1}$.
As before, the corresponding quantities in the \emph{calibrated}
solution will be indicated by an additional bar on top of the quantity, i.e., for example
$\bar \chi_i$ for the indicator function of the corresponding phases, $\bar{\vec{n}}_{i,i+1}$
for the corresponding normal, and so on.

\begin{figure}
	\begin{tikzpicture}[scale=2.5]
		\begin{scope}
			\clip (0,0) circle [radius=.8];

			\draw[thick] (0,0) arc (60:90:2);
			\draw[thick] (0,0) arc (180:205:2);
			\draw[thick] (0,0) arc (120:70:1);

		\end{scope}

			\node at (90:.25){${\bar{\Omega}}_i$};
			\node at (210:.4){${\bar{\Omega}}_j$};
			\node at (330:.4){${\bar{\Omega}}_k$};

			\node at (12:.9){${\bar{I}}_{k,i}$};
			\node at (155:.9){${\bar{I}}_{i,j}$};
			\node at (278:.9){${\bar{I}}_{j,l}$};
	\end{tikzpicture}
	\caption{Sketch of a triple junction. \label{fig:triple}}
\end{figure}

The first key step is to construct extensions $\xi_{i,i+1}$, $i=1,2,3$,
of the unit normal vector field $\bar{\vec{n}}_{i,i+1}$ of the \emph{calibrated} interfaces ${\bar{I}}_{i,i+1}$. As in the case of a single interface, the extensions
$\xi_{i,i+1}$ and the velocity field $B$ are constructed to have the following properties:
\begin{itemize}[leftmargin=0.7cm]
\item The time evolution of the vector fields $\xi_{i,i+1}$ is approximately described
by transport along the flow of the velocity field $B$. More precisely, for the vector field $B$ we have for $i=1,2,3$ that
\begin{align*}
\quad\quad\quad~
\partial_t \xi_{i,i+1} = -(B\cdot \nabla)\xi_{i,i+1} - (\nabla B)^\mathsf{T}\xi_{i,i+1}
+ O(\dist(\cdot,{\bar{I}}_{i,i+1})).
\end{align*}
\item On each interface ${\bar{I}}_{i,i+1}$, $i=1,2,3$, of the calibrated solution, the normal part of the velocity field $B$ must satisfy $\bar{\vec{n}}_{i,i+1}\cdot B = \bar H_{i,i+1} := -\nabla^{\tan} \cdot \bar{\vec{n}}_{i,i+1}$, where $\bar H_{i,i+1}$ is the scalar mean curvature of ${\bar{I}}_{i,i+1}$. We strengthen this identity to approximately
hold even away from the interface, in form of
\begin{align*}
\xi_{i,i+1}\cdot B = -\nabla \cdot \xi_{i,i+1} + O(\dist(\cdot,{\bar{I}}_{i,i+1}))
\quad\quad\text{for }i=1,2,3.
\end{align*}
\item The vector fields $\xi_{i,i+1}$ have at most unit length $|\xi_{i,i+1}|\leq 1$.
\item The length of the vector fields $\xi_{i,i+1}$ is advected with the flow of $B$ to higher order
\begin{align*}
\quad\quad\quad~
\partial_t |\xi_{i,i+1}|^2 = -(B\cdot \nabla)|\xi_{i,i+1}|^2
+ O\big(\dist^2(\cdot,{\bar{I}}_{i,i+1})\big)\quad\quad\text{for }i=1,2,3.
\end{align*}
\end{itemize}

The new aspect of a triple junction as opposed to a single interface is that one also has to extend the normal of an interface to locations where a different interface may be closer.
To this end, we turn to Herring's angle condition \eqref{HerringAngleCondition}, which in our case of equal surface tensions says
that the three interfaces must meet at the triple junction to form equal angles of $120^\circ$ each, and
require it to hold throughout the domain in the sense that
\begin{align}\label{eq:herringthrougout}
\sum_{i=1}^3 \xi_{i,i+1}(x,t)=0\quad\quad\text{for all }x,t.
\end{align}

Furthermore, note carefully that we only define a single extension $B$ of the velocity field, and that
$B$ is not necessarily a normal vector field on each interface ${\bar{I}}_{i,i+1}$:
Indeed, we expect the triple junction $p(t)$ to move according to $\frac{\mathrm{d}}{\mathrm{d}t} p = B(p(t),t)$, so that not
allowing for tangential components would pin the triple junction in space.
It turns out that in addition to Herring's angle condition, which we take to be of first order, we require higher-order compatibility conditions of the interfaces at the triple junction.
For instance, Definition~\ref{DefinitionStrongSolution}
of a strong solution requires the validity of the evolution law \eqref{MotionByMeanCurvature} also on the boundary points of ${\bar{I}}_{i,i+1}$ for $i=1,2,3$.
Hence, if we denote the tangent $\bar \tau_{i,i+1}:= J^{-1} \bar{\vec{n}}_{i,i_1}$ with $\smash{J:= \big(\begin{smallmatrix} 0 & -1 \\ 1 & 0	\end{smallmatrix}\big)}$ and $\alpha_{i,i+1}:= B \cdot \bar{\tau}_{i,i+1}$, we get $H_{i,i+1}\vec{\bar{n}}_{i,i+1} + \alpha_{i,i+1}\bar{\tau}_{i,i+1}
=H_{j,j+1}\vec{\bar{n}}_{j,j+1} + \alpha_{j,j+1}\bar{\tau}_{j,j+1}$
for all $i\neq j$ at the triple point.
Multiplying these identities with $\vec{\bar{n}}_{i,i+1}$, then varying over $j$ as well as making
use of Herring's angle condition implies the \emph{second-order compatibility condition} $H_{1,2}+H_{2,3}+H_{3,1}=0$,
see \cite{MantegazzaNovagaTortorelli}.

To construct the extensions $\xi_{i,i+1}$ of the normal vector fields
$\vec{\bar n}_{i,i+1}$, $i=1,2,3$, we first partition space into six wedge-shaped sets around the triple junction:
Three contain one strong interface each, while the remaining three wedges lie entirely within a single phase, see Figure~\ref{fig:triple_wedges}.
On the mixed phase wedges, we first extend the corresponding normal by an expansion ansatz, see Figure~\ref{fig:triple_extension}, and then define the remaining vector
fields to satisfy the identity \eqref{eq:herringthrougout} by $120^\circ$ rotations of the ansatz, see Figure~\ref{fig:triple_rotation}.
On the single phase wedges, we will interpolate between the competing definitions of the two adjacent mixed phase wedges.
\begin{figure}
	\centering
     \subcaptionbox{\label{fig:triple_wedges}}{
	  \centering
	  \begin{tikzpicture}[scale=3]
	  	\begin{scope}
			\clip (0,0) circle [radius=.8];
			\fill[pattern=north west lines,pattern color=blue,opacity=.2] (120:.8) -- (1,1.3) -- (1,-1)--  (300:.95) -- (120:.8);
			\fill[pattern=north east lines,pattern color=red,opacity=.2] (60:.8) -- (-1,1.3) -- (-1,-1)--  (240:.95) -- (60:.8);

			\foreach \i in {0,...,35}{
				\draw[opacity=.05] (-1,-\i*0.025) -- (1,-\i*0.025);
			};

		\draw[very thick,opacity=.6,color=red] (0,0) arc (60:90:2);
		\draw[very thick,opacity=.3] (0,0) arc (180:205:2);
		\draw[very thick,color=blue] (0,0) arc (120:70:1);

		\draw[color=red,opacity=.4] (0,0) -- (60:.8);
		\draw[color=blue,opacity=0.5] (0,0) -- (120:.8);
		\draw[opacity=.1]  (0,0) -- (180:1);
		\draw[color=red,opacity=.4] (0,0) -- (240:.95);
		\draw[color=blue,opacity=0.5] (0,0) -- (300:.95);
		\draw[opacity=.1] (0,0) -- (0:.85);

		\end{scope};

		\node at (10:.9){${\bar{I}}_{k,i}$};
		\node at (168:.9){${\bar{I}}_{i,j}$};
		\node at (290:.9){${\bar{I}}_{j,k}$};

		\node at (260:.9){$W_{j,k}$};
		\node at (330:.9){$W_{k}$};
		\node at (30:.9){$W_{k,i}$};
		\node at (90:.9){$W_{i}$};
		\node at (145:.9){$W_{i,j}$};
		\node at (210:.9){$W_{j}$};
	\end{tikzpicture}
     }
     \subcaptionbox{\label{fig:triple_extension}}{
	  \centering
	 	\begin{tikzpicture}[scale=3]
		\begin{scope}
			\clip (0,0) circle [radius=.8];
			\draw[thick,opacity=.6,color=red] (0,0) arc (60:90:2);
			\draw[thick,opacity=.3] (0,0) arc (180:205:2);
			\draw[thick,color=blue] (0,0) arc (120:70:1);

			\draw[color=red,opacity=.6] (0,0) -- (60:.8);
			\draw[color=blue] (0,0) -- (120:.8);
			\draw[opacity=.3]  (0,0) -- (180:1);
			\draw[color=red,opacity=.6] (0,0) -- (240:.95);
			\draw[color=blue] (0,0) -- (300:.95);
			\draw[opacity=.3] (0,0) -- (0:.85);

		\end{scope}

			\node at (40:.9){$W_{k,i}$};
			\node at (90:.9){$W_{i}$};
			\node at (135:.9){$W_{i,j}$};

			\foreach \i in {0,...,9}
			{
				\foreach \j in {-2,...,2}
				{
					\begin{scope}[shift={($(300:1)+(120 - 5*\i :1)+(120-5*\i : .13*\j)$) }]
						\draw[-{Latex[length=1mm]},color=blue]  (0,0) -- ($(120-5*\i - 10*\j - 1.7\j*\i:.1)$);
					\end{scope}
				}
			}

			\foreach \i in {0,...,8}
			{
				\foreach \j in {-2,...,2}
				{
					\begin{scope}[shift={($(240:2)+(60 + 2.7*\i :2) + (60+2.7*\i: .13*\j)$) }]
						\draw[-{Latex[length=1mm]},opacity=.6,color=red]  (0,0) -- ($(240+2.7*\i - 10*\j + 1.7*\j*\i  :.1)$);
					\end{scope}
				}
			}
	\end{tikzpicture}
     }
     \subcaptionbox{\label{fig:triple_rotation}}{
	  \centering
	 	\begin{tikzpicture}[scale=3]
	 	\begin{scope}
	 		\clip (0,0)  circle [radius=.8];

			\draw[thick,opacity=.6,color=red] (0,0) arc (60:90:2);
			\draw[thick,opacity=.3] (0,0) arc (180:205:2);
			\draw[thick,color=blue] (0,0) arc (120:70:1);

			\draw[color=red,opacity=.6] (0,0) -- (60:.8);
			\draw[color=blue] (0,0) -- (120:.8);
			\draw[opacity=.3]  (0,0) -- (180:1);
			\draw[color=red,opacity=.6] (0,0) -- (240:.95);
			\draw[color=blue] (0,0) -- (300:.95);
			\draw[opacity=.3] (0,0) -- (0:.85);
		\end{scope}

			\node at (40:.9){$W_{k,i}$};
			\node at (90:.9){$W_{i}$};
			\node at (135:.9){$W_{i,j}$};

			\foreach \i in {0,...,9}
			{
				\foreach \j in {-2,...,2}
				{
					\begin{scope}[shift={($(300:1)+(120 - 5*\i :1)+(120-5*\i : .13*\j)$) }]
						\draw[-{Latex[length=1mm]},color=blue]  (0,0) -- ($(120-5*\i - 10*\j - 1.7\j*\i:.1)$);
					\end{scope}
				}
			}

			\foreach \i in {0,...,8}
			{
				\foreach \j in {-2,...,2}
				{
					\begin{scope}[shift={($(240:2)+(60 + 2.7*\i :2) + (60+2.7*\i: .13*\j)$) }]
						\draw[-{Latex[length=1mm]},opacity=.6,color=red]  (0,0) -- ($(120+2.7*\i - 10*\j + 1.7*\j*\i  :.1)$);
					\end{scope}
				}
			}
	\end{tikzpicture}
     }
     \caption{a) The gray, horizontally hatched half space is $\mathbb{H}_{j,k}$, the region hatched in red from the bottom left to the top right is $\mathbb{H}_{i,j}$, and $\mathbb{H}_{k,i}$ is shown hatched in blue from the top left to the bottom right. The region simply hatched regions indicate the wedges $W_{i,j}$, $W_{j,k}$ and $W_{k,i}$ containing the interfaces ${\bar{I}}_{i,j}$, ${\bar{I}}_{j,k}$ and ${\bar{I}}_{k,i}$. The interpolation wedges $W_i$, $W_j$ and $W_k$ are shown as doubly hatched regions. b) Sketch of the initial extensions of $\vec{n}_{k,i}$ in blue on the right and $\vec{n}_{i,j}$ in red on the left, defined on $W_{k,i}$ and $W_{i,j}$, as well as the two respective neighboring interpolation wedges. c) The image shows the vector field $\vec{n}_{k,i}$ (in blue on the right) and the rotated vector field $R \vec{n}_{i,j}$ (in red on the left), where $R$ is the clockwise rotation by $120^\circ$.}
        \label{fig:triple_xi}
\end{figure}

All rigorous discussions of compatibility will be deferred to Section \ref{SectionLocalConstructionsTriod}, and we will only describe the initial extension procedure here.
Let us fix $i=1,2,3$. In fact, it is more instructive to first extend the velocity field $B$
in the wedge-shaped neighborhood of the interface ${\bar{I}}_{i,i+1}$.
To this end, we use the extension ansatz
\begin{align*}
	B := \bar{H}_{i,i+1} \vec{\bar n}_{i,i+1} + \alpha_{i,i+1} \bar \tau_{i,i+1} +\beta_{i,i+1} s_{i,i+1} \bar \tau_{i,i+1},
\end{align*}
where $\vec{\bar{n}}_{i,i+1}$ and $\bar \tau_{i,i+1}$ are extended to be constant in the $\vec{\bar{n}}_{i,i+1}$-direction, $s_{i,i+1}$ is the signed distance function to ${\bar{I}}_{i,i+1}$ with the sign convention $\nabla s_{i,i+1} = \bar{\vec{n}}_{i,i+1}$, and $ \alpha_{i,i+1}$ and $\beta_{i,i+1}$ are still to be determined.
All quantities except $B$ depend on the interface, we merely dropped the indices $i,i+1$ for ease of notation.
As $\frac{\mathrm{d}}{\mathrm{d}t} p (t)= B(p(t),t)$, it is reasonable that $\alpha_{i,i+1}(p(t),t) := \bar{\tau}_{i,i+1}(p(t),t)\cdot \frac{\mathrm{d}}{\mathrm{d}t} p(t)$ should be the tangential
velocity of $p$ at the triple junction.
It turns out to be convenient to extended $\alpha_{i,i+1}$ along the interface ${\bar{I}}_{i,i+1}$
by means of the \emph{ordinary} differential equation $(\bar{\tau}_{i,i+1}\cdot\nabla)\alpha_{i,i+1}=\bar{H}^2_{i,i+1}$. It can also be seen that
$\beta_{i,i+1}(x,t): = (\bar \tau_{i,i+1} \cdot \nabla ) {\bar{H}}_{i,i+1} + \alpha_{i,i+1} {\bar{H}}_{i,i+1}$
for $x\in {\bar{I}}_{i,i+1}(t)$ is a good candidate to make $B$ independent of $i$.
Indeed, the fact that the coefficient function $\beta_i(p(t),t)$ is independent of $i$ follows by differentiating Herring's angle condition in time, see \eqref{eq:betaIndependent}.
It therefore represents another second-order compatibility condition .
To define $\alpha_{i,i+1}$ and $\beta_{i,i+1}$ away from the interface, we once again require them to be constant in $\bar{\vec{n}}_{i,i+1}$-direction.

It turns out that as the extension $\xi=\xi_{i,i+1}(x,t)$ of $\vec{\bar n}_{i,i+1}$ one should take
\begin{equation}\label{xi3j_overview}
	\xi= \vec{\bar{n}} + \alpha s \bar \tau - \tfrac12 \alpha^2 s^2 \vec{\bar{n}} +
	\frac{1}{2}\beta s^2 \bar \tau,
\end{equation}
where the functions $\alpha=\alpha_{i,i+1}(x,t)$ and $\beta=\beta_{i,i+1}(x,t) $ are as above and we dropped the indices $i,i+1$ for ease of notation.
Note that in particular $\xi_{i,i+1}=\bar{\vec{n}}_{i,i+1}$ on the interface ${\bar{I}}_{i,i+1}$
and we allow for linear corrections of the tangential component as we move away from the interface,
but only for quadratic corrections of the normal component of $\xi$.

We then measure the error between the weak solution $\chi$ and the calibrated solution $\bar \chi$
by means of the relative entropy functional
\begin{align*}
E[\chi|\xi](t)
:=\sum_{i=1}^3 \int_{I_{i,i+1}(t)} (1 - \vec{n}_{i,i+1} \cdot \xi_{i,i+1}) \,\mathrm{d}\mathcal{H}^{1}.
\end{align*}
Let us use the abbreviation
$\sum_i = \sum_{i=1}^3$ for the summation over the three relevant indices.

As in the two-phase case, we would like to make use of just two ingredients to evaluate the time evolution of the relative entropy: the energy dissipation inequality for the weak solution in the sharp form
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} \sum_i \int_{I_{i,i+1}} 1 \,\mathrm{d}\mathcal{H}^{1} \leq -\sum_{i=1}^3 \int_{I_{i,i+1}} V_{i,i+1}^2 \,\mathrm{d}\mathcal{H}^{1},
\end{align*}
and the weak formulation of the evolution equation of the indicator functions $\chi_i$
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Rd} \chi_i \varphi \,\mathrm{d}x = \int_{\partial^* \Omega_i} V_i \varphi \,\,\mathrm{d}\mathcal{H}^{1}  +\int_{\Rd} \chi_i \partial_t \varphi \,\mathrm{d}x
\end{align*}
for compactly supported, smooth $\varphi$.
In order to make use of the latter equation, we have to rewrite the contributions $\int_{I_{i,i+1}} \vec{n}_{i,i+1} \cdot \xi_{i,i+1}(x,t) $ as a volume integral. It turns out that the annihilation condition $\sum_i \xi_{i,i+1}(x,t)=0$ enables us to rewrite $\xi_{i,i+1}$ as
\begin{align}\label{overview frame xi}
\xi_{i,i+1} = \xi_i-\xi_{i+1}
\end{align}
by defining the vector field $\xi_{i}$ as $\xi_{i} :=\frac{1}{3} (\xi_{i,i+1}-\xi_{i-1,i})$.
Combining \eqref{overview frame xi} with the symmetry $\vec{n}_{i,i+1}=-\frac{\mathrm{d}\nabla \chi_i}{\mathrm{d}|\nabla \chi_i|}=\frac{\mathrm{d}\nabla \chi_{i+1}}{\mathrm{d}|\nabla \chi_{i+1}|}$ and the decomposition $\partial^* \Omega_{i} = I_{i-1,i} \cup I_{i,i+1}$, we rewrite the second term in the relative entropy as
\begin{align*}
	-\sum_i \int_{I_{i,i+1}}   \vec{n}_{i,i+1} \cdot \xi_{i,i+1} \,\mathrm{d}\mathcal{H}^{1}
	&=
	\sum_i \bigg(\int_{I_{i,i+1}} \xi_i\cdot \mathrm{d}\nabla \chi_i+\int_{I_{i,i+1}} \xi_{i+1}\cdot \mathrm{d} \nabla \chi_{i+1}\bigg)
	\\&
	=
	\sum_i \int_{\partial^* \Omega_i} \xi_i\cdot \mathrm{d}\nabla \chi_i
	\\&
	= -\sum_i \int_{\Rd} \chi_i (\nabla \cdot \xi_i) \,\mathrm{d}x.
\end{align*}
This indeed enables us to evaluate the time evolution of the relative entropy as
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} E[\chi|\xi]
\leq&
-\sum_i \int_{I_{i,i+1}} V_{i,i+1}^2\,\mathrm{d}\mathcal{H}^{1}
\\&
-\sum_i  \int_{\partial^* \Omega_i} V_i (\nabla \cdot \xi_i)\,\mathrm{d}\mathcal{H}^{1}
+\sum_i \int_{\partial^* \Omega_i} \partial_t \xi_i \cdot \mathrm{d} \nabla \chi_i \,\mathrm{d}\mathcal{H}^{1}.
\end{align*}
Arguing analogously to the previous computation in reverse order---that is,
splitting the integrals into contributions $\partial^* \Omega_i\cap \partial^* \Omega_{i+1} = I_{i,i+1}$,
using \eqref{overview frame xi} and the definitions of $\vec{n}_{i,i+1}$ and $V_{i,i+1}$---we obtain
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t} E[\chi|\xi]
\leq
-\sum_i \int_{I_{i,i+1}} V_{i,i+1}^2\,\mathrm{d}\mathcal{H}^{1}
&-\sum_i  \int_{I_{i,i+1}} V_{i,i+1} (\nabla \cdot \xi_{i,i+1})\,\mathrm{d}\mathcal{H}^{1}
\\&
-\sum_i \int_{I_{i,i+1}} \partial_t \xi_{i,i+1} \cdot \vec{n}_{i,i+1}\,\mathrm{d}\mathcal{H}^{1}.
\end{align*}

Now we proceed as in the two-phase case in the previous section: The $\BV$ formulation of
mean curvature flow in this three-phase setting reads
\begin{align*}
&\sum_i \int_{I_{i,i+1}} V_{i,i+1} \vec{n}_{i,i+1} \cdot B \,\mathrm{d}\mathcal{H}^{1}
=- \sum_i \int_{I_{i,i+1}} (\operatorname{Id}-\vec{n}_{i,i+1}\otimes \vec{n}_{i,i+1}) : \nabla B\,\mathrm{d}\mathcal{H}^{1}.
\end{align*}
Following precisely the same algebraic manipulations as in the two-phase case we obtain
\begin{align*}
		\frac{\mathrm{d}}{\mathrm{d}t} &E[\chi | \xi]
	\\
	\leq& -\frac12 \sum_i \int_{I_{i,i+1}}
	\left(
	\left(
	V_{i,i+1}+\nabla \cdot \xi_{i,i+1}
	\right)^2
	+
	\left|
	V_{i,i+1}\vec{n}_{i,i+1} - \left(\xi_{i,i+1}\cdot B\right)\xi_{i,i+1}
	\right|^2
	\right)
	\,\mathrm{d}\mathcal{H}^{1}
	\\
	&+\frac12 \sum_i \int_{I_{i,i+1}}
	\left(
	\nabla \cdot \xi_{i,i+1}
	+
	\xi_{i,i+1} \cdot B
	\right)^2 \,\mathrm{d}\mathcal{H}^{1}
	\\&+ \sum_i \int_{I_{i,i+1}} \left(\nabla \cdot \xi_{i,i+1} \right)
	\left(\vec{n}_{i,i+1}\cdot \xi_{i,i+1} -1\right) \left(\xi_{i,i+1} \cdot B\right) \,\mathrm{d}\mathcal{H}^{1}
	\\
	&+\sum_i\int_{I_{i,i+1}} \left(\nabla \cdot \xi_{i,i+1} + V_{i,i+1} \right)
	\vec{n}_{i,i+1} \cdot  \left(\operatorname{Id} - \xi_{i,i+1} \otimes \xi_{i,i+1} \right) B\,\mathrm{d}\mathcal{H}^{1}
	 \\
	&+\sum_i\int_{I_{i,i+1}} (1-\vec{n}_{i,i+1}\cdot \xi_{i,i+1} ) \left(\nabla\cdot  B\right) \,\mathrm{d}\mathcal{H}^{1}
	\\&-\sum_i \int_{I_{i,i+1}}   \left(\vec{n}_{i,i+1}-\xi_{i,i+1}\right) \otimes
	\left(\vec{n}_{i,i+1}-\xi_{i,i+1}\right) \colon  \nabla  B \,\mathrm{d}\mathcal{H}^{1}
	\\
	&-\sum_i \int_{I_{i,i+1}} \left(\partial_t \xi_{i,i+1}
	+\left( B\cdot \nabla \right)\xi_{i,i+1}
	+ (\nabla B)^\mathsf{T} \xi_{i,i+1}\right)\cdot (\vec{n}_{i,i+1}-\xi_{i,i+1})\,\mathrm{d}\mathcal{H}^{1}
	\\
	&+ \sum_i\int_{I_{i,i+1}} \left(\partial_t \xi_{i,i+1} +\left( B\cdot \nabla \right)\xi_{i,i+1} \right)\cdot \xi_{i,i+1}\,\mathrm{d}\mathcal{H}^{1}.
\end{align*}
With this inequality at our disposal we can conclude as in the two-phase case.

\section{Stability of calibrated flows}
\label{SectionWeakStrongUniqueness}
This section is devoted to the proof of the stability properties of calibrated flows. In the next three subsections, we derive the relative
entropy inequality Proposition~\ref{PropositionRelativeEntropyInequality}, the stability estimate Proposition~\ref{PropositionUniqueness}, and the
volume error control Proposition~\ref{PropositionVolumeError}.

\subsection{Relative entropy inequality: Proof of  Proposition~\ref{PropositionRelativeEntropyInequality}}
\label{SectionDerivationRelativeEntropyInequality}
Consider a BV solution $\chi=(\chi_1,\ldots,\chi_P)$ of multiphase mean curvature flow in
the sense of Definition~\ref{DefinitionBVSolution} on some time interval $[0,T_{\BV})$, and let
$\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution of multiphase mean curvature flow in
the sense of Definition~\ref{DefinitionStrongSolution} on some time interval $[0,T_{\mathrm{strong}})$ with $T_{\mathrm{strong}}\leqT_{\BV}$.
Recall from \eqref{DefinitionRelativeEntropyFunctional} the definition of our relative entropy functional $E[\chi|\bar\chi]$.
The aim of this section is to provide the proof of the relative entropy inequality.

\begin{proof}[Proof of Proposition~\ref{PropositionRelativeEntropyInequality}]
In order to make use of the evolution equations \eqref{EvolutionPhasesBVSolution}
for the indicator functions $\chi_i$ of the BV solution,
we start by rewriting the interface error control of our relative entropy.
Using $\sigma_{i,j}\xi_{i,j}=\xi_i-\xi_j$ from Definition~\ref{DefinitionCalibrationGradientFlow}
of a gradient flow calibration, the symmetry relation $\vec{n}_{i,j}=-\vec{n}_{j,i}$,
the definition \eqref{UnitNormalsBVSolution} of the measure theoretic normal
as well as the representation of the energy \eqref{eq:energy},
we obtain by an application of the generalized divergence theorem
\begin{align}\label{eq:SurfaceToVolume}
\nonumber
&\sum_{i,j=1,i\neq j}^P \sigma_{i,j} \int_{I_{i,j}(T)}
1-\xi_{i,j}(\cdot,T) \cdot \vec{n}_{i,j}(\cdot,T)
\,\mathrm{d}\mathcal{H}^{d-1}
\\&\nonumber
= E[\chi(\cdot,T)] - \sum_{i,j=1,i\neq j}^P \int_{I_{i,j}(T)}
(\xi_i(\cdot,T){-}\xi_j(\cdot,T))\cdot \vec{n}_{i,j}(\cdot,T)
\,\mathrm{d}\mathcal{H}^{d-1}
\\&\nonumber
= E[\chi(\cdot,T)]
+ \sum_{i=1}^P\sum_{j=1,j\neq i}^P\int_{I_{i,j}(T)}
\xi_i(\cdot,T)\cdot \frac{\nabla\chi_i(\cdot,T)}{|\nabla\chi_i(\cdot,T)|}
\,\mathrm{d}\mathcal{H}^{d-1}
\\&~~~\nonumber
+ \sum_{j=1}^P\sum_{i=1,i\neq j}^P\int_{I_{i,j}(T)}
\xi_j(\cdot,T)\cdot \frac{\nabla\chi_j(\cdot,T)}{|\nabla\chi_j(\cdot,T)|}
\,\mathrm{d}\mathcal{H}^{d-1}
\\&\nonumber
= E[\chi(\cdot,T)] + 2\sum_{i=1}^P \int_{\Rd}
\xi_i(\cdot,T)\cdot \frac{\nabla\chi_i(\cdot,T)}{|\nabla\chi_i(\cdot,T)|}
\,\mathrm{d}|\nabla\chi_i(\cdot,T)|
\\&
= E[\chi(\cdot,T)] - 2\sum_{i=1}^P \int_{\Rd} \chi_i(\cdot,T)
(\nabla\cdot\xi_i(\cdot,T))\,\mathrm{d}x.
\end{align}
This enables us to compute by the sharp energy dissipation inequality \eqref{EnergyDissipationInequalityBVSolution},
the evolution equations \eqref{EvolutionPhasesBVSolution} for the indicator functions $\chi_i$
of the BV solution, and definition \eqref{NormalVelocitiesBVSolution} of the velocities $V_{i,j}$
for almost every $T\in (0,T_{\mathrm{strong}})$
\begin{align*}
&E_{}[\chi|\xi](T)
\\&
\leq E[\chi(\cdot,0)] - 2\sum_{i=1}^P \int_{\Rd} \chi_{0,i}
(\nabla\cdot\xi_i(\cdot,0))\,\mathrm{d}x
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
|V_{i,j}|^2\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
-2\sum_{i=1}^P \int_0^T\int_{\Rd}
\chi_i\partial_t(\nabla\cdot\xi_i)\,\mathrm{d}x\,\mathrm{d}t
-2\sum_{i=1}^P \int_0^T\int_{\Rd}
V_i(\nabla\cdot\xi_i)\,\mathrm{d}|\nabla\chi_i|\,\mathrm{d}t.
\end{align*}
The first two terms combine to $E_{\mathrm{interface}}[\chi|\bar\chi](0)$ using \eqref{eq:SurfaceToVolume}
in reverse order. We aim to rewrite the latter two terms back to surface integrals over the interfaces as well.
To this end, we argue analogously to the computation in \eqref{eq:SurfaceToVolume} but now in reverse order.
Using first the generalized divergence theorem, then splitting the integrals over the reduced boundaries of the phases
into contributions over the interfaces $I_{i,j}=\partial^* \Omega_i\cap \partial^* \Omega_{j}$
by means of $\sigma_{i,j}\xi_{i,j}=\xi_{i}-\xi_{j}$ from Definition~\ref{DefinitionCalibrationGradientFlow}
of a gradient flow calibration we obtain
\begin{align*}
-2\sum_{i=1}^P \int_0^T\int_{\Rd}\chi_i\partial_t(\nabla\cdot\xi_i)\,\mathrm{d}x\,\mathrm{d}t
&=2\sum_{i=1}^P \int_0^T\int_{\Rd}\frac{\nabla\chi_i}{|\nabla\chi_i|}\cdot\partial_t\xi_i
\,\mathrm{d}|\nabla\chi_i|\,\mathrm{d}t
\\&
=\sum_{i=1}^P\sum_{j=1,j\neq i}^P\int_0^T\int_{I_{i,j}(t)}
\frac{\nabla\chi_i}{|\nabla\chi_i|}\cdot\partial_t\xi_i
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~
+\sum_{j=1}^P\sum_{i=1,i\neq j}^P\int_0^T\int_{I_{i,j}(t)}
\frac{\nabla\chi_j}{|\nabla\chi_j|}\cdot\partial_t\xi_j
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
\stackrel{\eqref{UnitNormalsBVSolution}}{=}
-\sum_{i,j=1,i\neq j}^P\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot\partial_t(\xi_i-\xi_j)\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
=-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot\partial_t\xi_{i,j}\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t.
\end{align*}
The term incorporating the normal velocities is treated similarly.
In addition to the above ingredients, i.e., $\sigma_{i,j}\xi_{i,j}=\xi_{i}-\xi_{j}$
from Definition~\ref{DefinitionCalibrationGradientFlow} of a gradient flow calibration and
splitting the integrals over the reduced boundaries of the phases
into contributions over the interfaces $I_{i,j}=\partial^* \Omega_i\cap \partial^* \Omega_{j}$,
we also use that $V_{i,j}=-V_{j,i}$ on ${\bar{I}}_{i,j}$ together with definition
\eqref{NormalVelocitiesBVSolution} to compute
\begin{align*}
-2\sum_{i=1}^P \int_0^T\int_{\Rd}
V_i(\nabla\cdot\xi_i)\,\mathrm{d}|\nabla\chi_i|\,\mathrm{d}t
&=-\sum_{i=1}^P\sum_{j=1,j\neq i}^P\int_0^T\int_{I_{i,j}(t)}
V_{i,j}(\nabla\cdot\xi_i)\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~
+\sum_{j=1}^P\sum_{i=1,i\neq j}^P\int_0^T\int_{I_{i,j}(t)}
V_{i,j}(\nabla\cdot\xi_j)\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
=-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
V_{i,j}(\nabla\cdot\xi_{i,j})\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t.
\end{align*}
Combining the last two identities, we obtain for almost every $T\in (0,T_{\mathrm{strong}})$
\begin{align*}
&E_{}[\chi|\xi](T)
\\&
\leq E_{}[\chi|\xi](0)
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
|V_{i,j}|^2\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot\partial_t\xi_{i,j}\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
V_{i,j}(\nabla\cdot\xi_{i,j})\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t.
\end{align*}
For the next step, we use the vector field $B$
as a test function in the BV formulation of mean curvature flow \eqref{BVFormulationMeanCurvature}. Adding the
resulting equation to the previous inequality, observing in the process that
$V_i\frac{\nabla\chi_i}{|\nabla\chi_i|}=-V_{i,j}\vec{n}_{i,j}$ on $I_{i,j}$
due to \eqref{UnitNormalsBVSolution} and \eqref{NormalVelocitiesBVSolution},
as well as decomposing $B=(\mathrm{Id}{-}\xi_{i,j}\otimes\xi_{i,j})B
+(B\cdot\xi_{i,j})\xi_{i,j}$, we obtain
\begin{align}\label{DerivationRelEntropyInequ1}
\nonumber
&E_{}[\chi|\xi](T)
\\&
\leq E_{}[\chi|\xi](0)
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
|V_{i,j}|^2\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(B\cdot\xi_{i,j})\xi_{i,j}\cdot V_{i,j}\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
V_{i,j}(\nabla\cdot\xi_{i,j})\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\mathrm{Id}{-}\xi_{i,j}\otimes\xi_{i,j})B\cdot V_{i,j}\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\nabla\cdot B)\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\otimes\vec{n}_{i,j}:\nabla B\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot\partial_t\xi_{i,j}\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t,
\end{align}
which holds for almost every $T\in (0,T_{\mathrm{strong}})$.
In order to obtain the dissipation term on the left hand side of the
relative entropy inequality \eqref{RelativeEntropyInequality}, we complete
the squares yielding for almost every $T\in (0,T_{\mathrm{strong}})$
\begin{align}\label{DerivationRelEntropyInequ2}
\nonumber
&-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
|V_{i,j}|^2\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(B\cdot\xi_{i,j})\xi_{i,j}\cdot V_{i,j}\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
V_{i,j}(\nabla\cdot\xi_{i,j})\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\nonumber
= -\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}\Big(
\frac{1}{2}|V_{i,j}{+}\nabla\cdot\xi_{i,j}|^2
+\frac{1}{2}|V_{i,j}\vec{n}_{i,j}{-}(B\cdot\xi_{i,j})\xi_{i,j}|^2\Big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
+ \sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\Big(\frac{1}{2}|\nabla\cdot\xi_{i,j}|^2+\frac{1}{2}|(B\cdot\xi_{i,j})\xi_{i,j}|^2\Big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t.
\end{align}
Furthermore, on the one hand, adding and subtracting $(B\cdot\nabla)\xi_{i,j}+(\nabla B)^\mathsf{T}\xi_{i,j}$ yields
\begin{align}\label{DerivationRelEntropyInequ3}
\nonumber
&\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\nabla\cdot B)\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\nonumber
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\otimes\vec{n}_{i,j}:\nabla B\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\nonumber
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot\partial_t\xi_{i,j}\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
= \sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\nabla\cdot B)\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\vec{n}_{i,j}-\xi_{i,j})\cdot(\vec{n}_{i,j}\cdot\nabla)B\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\big((B\cdot\nabla)\xi_{i,j}\big)\cdot\vec{n}_{i,j}\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\big(\partial_t\xi_{i,j}{+}(B\cdot\nabla)\xi_{i,j}{+}(\nabla B)^\mathsf{T}\xi_{i,j}\big)\cdot\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\end{align}
for almost every $T\in (0,T_{\mathrm{strong}})$. On the other hand, we may exploit symmetry to obtain (relying again on the by now routine fact that one can
switch back and forth between certain volume integrals and surface integrals over the individual interfaces
by means of $\sigma_{i,j}\xi_{i,j}=\xi_i-\xi_j$ from Definition~\ref{DefinitionCalibrationGradientFlow}
of a gradient flow calibration, the symmetry relation $\vec{n}_{i,j}=-\vec{n}_{j,i}$ and
the definition \eqref{UnitNormalsBVSolution})
\begin{align*}
&\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot\big(\nabla\cdot(B\otimes\xi_{i,j})\big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
=\sum_{i,j=1,i\neq j}^{P}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot\big(\nabla\cdot(B\otimes(\xi_{i}{-}\xi_j))\big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
=-2\sum_{i=1}^P\int_0^T\int_{\Rd}\frac{\nabla\chi_i}{|\nabla\chi_i|}
\cdot\big(\nabla\cdot(B\otimes\xi_{i})\big)\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
=2\sum_{i=1}^P\int_0^T\int_{\Rd}\chi_i\nabla\cdot
\big(\nabla\cdot(B\otimes\xi_{i})\big)\,\mathrm{d}x\,\mathrm{d}t
\\&
=2\sum_{i=1}^P\int_0^T\int_{\Rd}\chi_i\nabla\cdot
\big(\nabla\cdot(\xi_{i}\otimes B)\big)\,\mathrm{d}x\,\mathrm{d}t
\\&
=\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot\big(\nabla\cdot(\xi_{i,j}\otimes B)\big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t.
\end{align*}
Because of this identity, we can now compute
\begin{align}
\label{DerivationRelEntropyInequ4}
\nonumber
0 &= \sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot\big(\nabla\cdot(B\otimes\xi_{i,j}-\xi_{i,j}\otimes B)\big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
= \sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\nabla\cdot\xi_{i,j})B\cdot\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
+ \sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot(\xi_{i,j}\cdot\nabla) B
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
- \sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\cdot(B\cdot\nabla)\xi_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
- \sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\nabla\cdot B)\xi_{i,j}\cdot\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t.
\end{align}
Making use of the identities \eqref{DerivationRelEntropyInequ2} and \eqref{DerivationRelEntropyInequ3}
in the inequality \eqref{DerivationRelEntropyInequ1} as well as adding \eqref{DerivationRelEntropyInequ4}
to the right hand side of \eqref{DerivationRelEntropyInequ1}, we arrive at the following bound for
the time evolution of the interface error control of our relative entropy functional
\begin{align}
\nonumber
&E_{}[\chi|\xi](T)
\\&\nonumber
+\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}\Big(
\frac{1}{2}|V_{i,j}{+}\nabla\cdot\xi_{i,j}|^2
+\frac{1}{2}|V_{i,j}\vec{n}_{i,j}{-}(B\cdot\xi_{i,j})\xi_{i,j}|^2\Big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\label{eq:prefinalRelEntropyInequality}
\leq E_{}[\chi|\xi](0)
\\&~~~\nonumber
+ \sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\Big(\frac{1}{2}|\nabla\cdot\xi_{i,j}|^2+\frac{1}{2}|(B\cdot\xi_{i,j})\xi_{i,j}|^2\Big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
+\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\nabla\cdot\xi_{i,j})B\cdot\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\mathrm{Id}{-}\xi_{i,j}\otimes\xi_{i,j})B\cdot V_{i,j}\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
+\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\nabla\cdot B)(1-\xi_{i,j}\cdot\vec{n}_{i,j})
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\vec{n}_{i,j}-\xi_{i,j})\otimes \vec{n}_{i,j}:\nabla B\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\otimes \xi_{i,j}:\nabla B\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\big(\partial_t\xi_{i,j}{+}(B\cdot\nabla)\xi_{i,j}{+}(\nabla B)^\mathsf{T}\xi_{i,j}\big)\cdot\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t,
\end{align}
which is valid for almost every $T\in (0,T_{\mathrm{strong}})$. Completing squares and adding zero
yields for the second, third and fourth term on the right hand side of \eqref{eq:prefinalRelEntropyInequality}
\begin{align}
\nonumber
&\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\Big(\frac{1}{2}|\nabla\cdot\xi_{i,j}|^2+\frac{1}{2}|(B\cdot\xi_{i,j})\xi_{i,j}|^2\Big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\nonumber
+\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\nabla\cdot\xi_{i,j})B\cdot\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\mathrm{Id}{-}\xi_{i,j}\otimes\xi_{i,j})B\cdot V_{i,j}\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\label{eq:prefinalRelEntropyInequality2}
=\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}|(\nabla\cdot\xi_{i,j})+B\cdot\xi_{i,j}|^2
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}|B\cdot\xi_{i,j}|^2(1-|\xi_{i,j}|^2)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\mathrm{Id}{-}\xi_{i,j}\otimes\xi_{i,j})B\cdot
(V_{i,j}+\nabla\cdot\xi_{i,j})\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(1-\vec{n}_{i,j}\cdot\xi_{i,j})(\nabla\cdot\xi_{i,j})(B\cdot\xi_{i,j})
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t.
\end{align}
Adding zero in the last term on the right hand side of \eqref{eq:prefinalRelEntropyInequality}
in order to generate the transport equation for the length of the vector fields $\xi_{i,j}$, we
observe that the last three terms on the right hand side of \eqref{eq:prefinalRelEntropyInequality}
combine to
\begin{align}
\nonumber
&-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\vec{n}_{i,j}-\xi_{i,j})\otimes \vec{n}_{i,j}:\nabla B\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\nonumber
+\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\vec{n}_{i,j}\otimes \xi_{i,j}:\nabla B\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\nonumber
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\big(\partial_t\xi_{i,j}{+}(B\cdot\nabla)\xi_{i,j}{+}(\nabla B)^\mathsf{T}\xi_{i,j}\big)\cdot\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\label{eq:prefinalRelEntropyInequality3}
=-\sum_{i,j=1,i\neq j}^P\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
(\vec{n}_{i,j}-\xi_{i,j})\otimes (\vec{n}_{i,j}-\xi_{i,j}):\nabla B\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\big(\partial_t\xi_{i,j}{+}(B\cdot\nabla)\xi_{i,j}{+}(\nabla B)^\mathsf{T}\xi_{i,j}\big)
\cdot(\vec{n}_{i,j}-\xi_{i,j})\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~\nonumber
-\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}\big(\partial_t|\xi_{i,j}|^2{+}(B\cdot\nabla)|\xi_{i,j}|^2\big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t.
\end{align}
Employing the notation of Proposition~\ref{PropositionRelativeEntropyInequality}
as well as using \eqref{eq:prefinalRelEntropyInequality2} and \eqref{eq:prefinalRelEntropyInequality3}
in \eqref{eq:prefinalRelEntropyInequality}, we deduce that the right hand side
of \eqref{eq:prefinalRelEntropyInequality} indeed reduces to
\begin{align*}
&E_{}[\chi|\xi](T)
\\&\nonumber
+\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}\Big(
\frac{1}{2}|V_{i,j}{+}\nabla\cdot\xi_{i,j}|^2
+\frac{1}{2}|V_{i,j}\vec{n}_{i,j}{-}(B\cdot\xi_{i,j})\xi_{i,j}|^2\Big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&
\leq E_{}[\chi|\xi](0)
+R_{\mathrm{dt}}+R_{\mathrm{dissip}},
\end{align*}
which is valid for almost every $T\in (0,T_{\mathrm{strong}})$.
This concludes the proof of \eqref{RelativeEntropyInequality}.
\end{proof}

\subsection{Quantitative inclusion principle: Proof of Proposition~\ref{PropositionUniqueness}}\label{SectionProofMainResult}

We now prove
 the inclusion principle stating that interfaces of any $\BV$ solution must be contained in the corresponding interfaces of a calibrated flow, provided both start with the same initial data.

\begin{proof}[Proof of Proposition~\ref{PropositionUniqueness}]

\emph{Step 1: The stability estimate \eqref{StabilityEstimate}.}
The starting point is the estimate on the evolution of the interface error functional \eqref{DefinitionRelativeEntropy} from Proposition~\ref{PropositionRelativeEntropyInequality}.
In the following, we estimate the terms appearing on
the right hand side one-by-one. Due to \eqref{TransportEquationXi}, \eqref{LengthConservation}, as well as the trivial relation
\begin{align}\label{eq:L2excess}
|\vec{n}_{i,j}{-}\xi_{i,j}|^2\leq 2(1-\vec{n}_{i,j}\cdot\xi_{i,j})
\end{align}
(which follows by $|\xi_{i,j}|\leq 1$),
we immediately deduce
\begin{align}\label{BoundRdt}
|R_{\mathrm{dt}}| \leq C\int_0^T E_{}[\chi|\xi](t)\,\mathrm{d}t.
\end{align}
Making use of the simple estimate $1{-}|\xi_{i,j}|^2\leq 2(1-|\xi_{i,j}|)\leq 2(1-\vec{n}\cdot\xi_{i,j})$
and again the bound \eqref{eq:L2excess}, we also obtain
\begin{align*}
|R_{\mathrm{dissip}}|
&\leq
\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}|(\nabla\cdot\xi_{i,j})+B\cdot\xi_{i,j}|^2
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
+\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
(\mathrm{Id}{-}\xi_{i,j}\otimes\xi_{i,j})B\cdot(V_{i,j}{+}\nabla\cdot\xi_{i,j})\vec{n}_{i,j}
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
+C\int_0^T E_{}[\chi|\xi](t)\,\mathrm{d}t
\\&
=: I + II + C\int_0^T E_{}[\chi|\xi](t)\,\mathrm{d}t.
\end{align*}
By means of \eqref{Dissip}, we may directly estimate
\begin{align*}
|I| \leq C\int_0^T E_{}[\chi|\xi](t)\,\mathrm{d}t.
\end{align*}
Furthermore, by an application of H\"older's and Young's inequality we deduce
\begin{align*}
|II| &= \bigg|\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}
\int_0^T\int_{I_{i,j}(t)}
(\mathrm{Id}{-}\xi_{i,j}\otimes\xi_{i,j})B\cdot(V_{i,j}{+}\nabla\cdot\xi_{i,j})
(\vec{n}_{i,j}{-}\xi_{i,j})\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t\bigg|
\\&
\leq \delta\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}(V_{i,j}{+}\nabla\cdot\xi_{i,j})^2\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~
+C\delta^{-1}\int_0^T E_{}[\chi|\xi](t)\,\mathrm{d}t,
\end{align*}
uniformly over all $\delta\in (0,1)$. Hence, we get the bound
\begin{align}\label{BoundRdissip}
|R_{\mathrm{dissip}}|
&\leq
\delta\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}(V_{i,j}{+}\nabla\cdot\xi_{i,j})^2\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&~~~~\nonumber
+C\delta^{-1}\int_0^T E_{}[\chi|\xi](t)\,\mathrm{d}t.
\end{align}

Plugging in the bounds from \eqref{BoundRdt} and \eqref{BoundRdissip}
into the relative entropy inequality from Proposition~\ref{PropositionRelativeEntropyInequality},
and then choosing $\delta\in (0,1)$ sufficiently small in order to absorb the first right-hand side term, we therefore get constants $C>0$ and $c>0$
such that the estimate
\begin{align}
\label{eq:postProcessedInterfaceError}
&E[\chi|\xi](T)
\\&\nonumber
+ c\sum_{i,j=1,i\neq j}^{P}
\int_0^T\int_{I_{i,j}(t)}\Big(\frac{1}{2}(V_{i,j}{+}\nabla\cdot\xi_{i,j})^2
+\frac{1}{2}|V_{i,j}\vec{n}_{i,j}{-}(B\cdot\xi_{i,j})\xi_{i,j}|^2\Big)
\,\mathrm{d}\mathcal{H}^{d-1}\,\mathrm{d}t
\\&\nonumber
\leq C \int_0^T E[\chi|\xi](t)\,\mathrm{d}t
\end{align}
holds true for almost every $T\in (0,T_{\mathrm{strong}})$. By an application of Gronwall's lemma,
the asserted stability estimate \eqref{StabilityEstimate} from Proposition~\ref{PropositionUniqueness} follows.

\emph{Step 3: Weak-strong comparison.}
In the case of coinciding initial conditions, i.e.\ $E[\chi|\xi](0)=0$, the stability estimate \eqref{StabilityEstimate} entails $E[\chi|\xi]=0$ for almost every $t\in [0,T_{\mathrm{strong}})$. From
this and \eqref{LengthControlXi}, it immediately follows that $I_{i,j}(t)\subset {\bar{I}}_{i,j}(t)$ holds up to an $\mathcal{H}^{d-1}$-null set
for almost every $t\in [0,T_{\mathrm{strong}})$.
This proves the weak-strong uniqueness principle for $\BV$~solutions of multiphase mean curvature flow in the plane.
\end{proof}

\subsection{Weighted volume error control: Proof of Proposition~\ref{PropositionVolumeError}}
\label{SectionVolumeError}
Unfortunately, the interface error control provided by the functional
from \eqref{DefinitionRelativeEntropyFunctional}
suffers from a lack of coercivity in the case of vanishing interface length
for a $\BV$ solution. It is for this reason that our relative
entropy functional not only consists of the interface error control but
also of a lower-order term $E_{\mathrm{volume}}[\chi|\bar\chi]$,
which provides an error control in the volume occupied by the grains
of the strong solution (and weighted by the distance to the respective grain boundary).
The aim of this section is to derive the main ingredient for the stability estimate \eqref{StabilityEstimateVolume},
and therefore Proposition~\ref{PropositionVolumeError}.
We start our discussion with the following pendant of Proposition~\ref{PropositionRelativeEntropyInequality}.
In analogy to Proposition~\ref{PropositionRelativeEntropyInequality}, we will only
require minimal assumptions on the weight functions defining the functional $E_{\mathrm{volume}}[\chi|\bar\chi]$.
The construction of such weight functions with sufficiently good coercivity properties is
deferred to the subsequent result.

\begin{lemma}
\label{relativeEntropyVolume}
Let $d=2$, $P\geq 2$ be an integer and $\sigma\in\Rd[P\times P]$ be an admissible matrix of surface tensions
in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions}.
Let $\chi=(\chi_1,\ldots,\chi_P)$ be a BV solution of multiphase mean curvature flow in
the sense of Definition~\ref{DefinitionBVSolution} on some time interval $[0,T_{\BV})$, and let
$\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution of multiphase mean curvature flow in
the sense of Definition~\ref{DefinitionStrongSolution} on some time interval $[0,T_{\mathrm{strong}})$ with $T_{\mathrm{strong}}\leqT_{\BV}$.
Recall from \eqref{UnitNormalsBVSolution} resp.\ \eqref{NormalVelocitiesBVSolution} the definitions of the
(measure-theoretic) unit normal vectors $\vec{n}_{i,j}$ resp.\ of the normal velocities $V_{i,j}$ of a BV solution.

Let $0<r<R$ be two radii such that $\bigcup_{i,j=1,i\neq j}^P {\bar{I}}_{i,j}(t) + B_{2r}(0)$
is compactly supported in $B_R(0)$ for all times $t\in [0,T_{\mathrm{strong}})$. Let then $(\vartheta_i)_{i=1,\ldots,P}$ be a family of functions satisfying
\begin{align*}
\vartheta_i\in L^\infty([0,T_{\mathrm{strong}});W^{1,\infty}(\Rd[2]))\cap
W^{1,\infty}([0,T_{\mathrm{strong}});L^\infty(\Rd[2])),
\end{align*}
$\supp\vartheta_i\subset B_{2R}(0)$,
$\vartheta_i\equiv 0$ on the phase boundary $\bigcup_{j=1}^P {\bar{I}}_{i,j}$
of the $i$-th phase $\{\bar\chi_i=1\}$ of the strong solution,
$\vartheta_i\leq 0$ in $\{\bar\chi_i=1\}$ and
$\vartheta_i\geq 0$ in $\{\bar\chi_i=0\}$.
Define then the weighted volume error control
\begin{align}
\label{volumeErrorFunctional}
E_{\mathrm{volume}}[\chi|\bar\chi](t)
:=
\sum_{i=1}^P
\int_{\Rd[2]}|\chi_i(\cdot,t){-}\bar\chi_i(\cdot,t)||\vartheta_i(\cdot,t)| \,\mathrm{d}x.
\end{align}
Moreover, let $(\xi_{i,j})_{i\neq j\in\{1,\ldots,P\}}$ be a family
of compactly supported vector fields such that
\begin{align*}
\xi_{i,j} \in L^\infty([0,T_{\mathrm{strong}});W^{2,\infty}(\Rd[2];\Rd[2]))\cap
W^{1,\infty}([0,T_{\mathrm{strong}});C^0(\Rd[2];\Rd[2])),
\end{align*}
and let
\begin{align*}
B\in L^\infty([0,T_{\mathrm{strong}});W^{1,\infty}(\Rd[2];\Rd[2]))
\end{align*}
be an arbitrary compactly supported vector field.

Then the weighted volume error control is subject to the following identity
\begin{align}
\label{RelativeEntropyInequalityVolumeControl}
E_{\mathrm{volume}}[\chi|\bar\chi](T)
= E_{\mathrm{volume}}[\chi|\bar\chi](0) + R_{\mathrm{volume}}.
\end{align}
for almost every $T\in (0,T_{\mathrm{strong}})$. Here, we made use of the abbreviation
\begin{align*}
R_{\mathrm{volume}} &:=
-\sum_{i=1,i\neq j}^{P}\int_0^T\int_{I_{i,j}(t)}
\vartheta_i(B\cdot\xi_{i,j}-V_{i,j})\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t
\\&~~~
-\sum_{i=1,i\neq j}^{P}\int_0^T\int_{I_{i,j}(t)}
\vartheta_iB\cdot(\vec{n}_{i,j}-\xi_{i,j})\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t
\\&~~~
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i-\bar\chi_i)
\vartheta_i(\nabla\cdot B)\,\mathrm{d}x\,\mathrm{d}t
\\&~~~
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i-\bar\chi_i)
(\partial_t\vartheta_i+(B\cdot\nabla)\vartheta_i)\,\mathrm{d}x\,\mathrm{d}t.
\end{align*}
\end{lemma}

\begin{proof}
We first observe that $E_{\mathrm{volume}}[\chi|\bar\chi]<\infty$
because of $\supp\vartheta_i\subset B_{2R}(0)$. To compute the time evolution,
note that the sign condition on $\vartheta_i$ is precisely what is needed to have
\begin{align*}
E_{\mathrm{volume}}[\chi|\bar\chi](T) =
\sum_{i=1}^{P}\int_{\Rd[2]}(\chi_i(\cdot,T){-}\bar\chi_i(\cdot,T))\vartheta_i(\cdot,T)\,\mathrm{d}x.
\end{align*}
Hence, we may make use of the evolution equations \eqref{EvolutionPhasesBVSolution}
for the indicator functions $\chi_i$ of the BV solution as well as the
ones for the indicator functions $\bar\chi_i$ of the strong solution, yielding
for almost every $T\in (0,T_{\mathrm{strong}})$
\begin{align*}
&E_{\mathrm{volume}}[\chi|\bar\chi](T)
\\&
= E_{\mathrm{volume}}[\chi|\bar\chi](0)
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)\partial_t\vartheta_i\,\mathrm{d}x\,\mathrm{d}t
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]} V_i\vartheta_i\,\mathrm{d}|\nabla\chi_i|\,\mathrm{d}t.
\end{align*}
Here we have used that $\vartheta_i|\nabla\bar\chi_i|\equiv 0$, since $\vartheta_i$ vanishes
on $\partial\{\bar\chi_i=1\}$. We next use \eqref{NormalVelocitiesBVSolution} and rewrite
\begin{align*}
\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]} V_i\vartheta_i\,\mathrm{d}|\nabla\chi_i|\,\mathrm{d}t
=\sum_{i=1,i\neq j}^{P}\int_0^T\int_{I_{i,j}(t)} \vartheta_iV_{i,j}\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t.
\end{align*}
Furthermore, by adding and subtracting $(B\cdot\nabla)\vartheta_i$,
an integration by parts, $\vartheta_i\equiv 0$ on $\partial\{\bar\chi_i=1\}$,
and the definition \eqref{UnitNormalsBVSolution} of the measure theoretic normal, we obtain
\begin{align*}
&\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)\partial_t\vartheta_i\,\mathrm{d}x\,\mathrm{d}t
\\&
= -\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)(B\cdot\nabla)\vartheta_i\,\mathrm{d}x\,\mathrm{d}t
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)(\partial_t\vartheta_i{+}(B\cdot\nabla)\vartheta_i)\,\mathrm{d}x\,\mathrm{d}t
\\&
=-\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)\nabla\cdot(\vartheta_iB)\,\mathrm{d}x\,\mathrm{d}t
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)\vartheta_i(\nabla\cdot B)\,\mathrm{d}x\,\mathrm{d}t
\\&~~~
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)(\partial_t\vartheta_i{+}(B\cdot\nabla)\vartheta_i)\,\mathrm{d}x\,\mathrm{d}t
\\&
=\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}
\frac{\nabla\chi_i}{|\nabla\chi_i|}\cdot\vartheta_iB
\,\mathrm{d}|\nabla\chi_i|\,\mathrm{d}t
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)\vartheta_i(\nabla\cdot B)\,\mathrm{d}x\,\mathrm{d}t
\\&~~~
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)(\partial_t\vartheta_i{+}(B\cdot\nabla)\vartheta_i)\,\mathrm{d}x\,\mathrm{d}t
\\&
=-\sum_{i=1,i\neq j}^{P}\int_0^T\int_{I_{i,j}(t)}
\vartheta_iB\cdot\xi_{i,j}\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t
\\&~~~
-\sum_{i=1,i\neq j}^{P}\int_0^T\int_{I_{i,j}(t)}
\vartheta_iB\cdot(\vec{n}_{i,j}{-}\xi_{i,j})\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t
\\&~~~
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)\vartheta_i(\nabla\cdot B)\,\mathrm{d}x\,\mathrm{d}t
\\&~~~
+\sum_{i=1}^{P}\int_0^T\int_{\Rd[2]}(\chi_i{-}\bar\chi_i)(\partial_t\vartheta_i{+}(B\cdot\nabla)\vartheta_i)\,\mathrm{d}x\,\mathrm{d}t
\end{align*}
for almost every $T\in (0,T_{\mathrm{strong}})$. All in all, we therefore get for almost every $T\in (0,T_{\mathrm{strong}})$ the identity
\eqref{RelativeEntropyInequalityVolumeControl} as asserted.
\end{proof}

In order to infer the stability estimate \eqref{StabilityEstimateVolume} based on
the identity \eqref{RelativeEntropyInequalityVolumeControl}, we have to construct
--- next to a gradient flow calibration --- a family of
suitable weight functions $\vartheta_i$ with sufficiently good coercivity properties.
As the following result shows, the existence of a strong solution always guarantees this.

\begin{lemma}
\label{LemmaWeightedVolumeControl}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution of multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution}
on some time interval $[0,T_{\mathrm{strong}})$. Let $\widetilde r_c>0$ be the localization
parameter from Lemma~\ref{LemmaPartitionOfUnity}. Choose $R=R(\bar\chi)>0$
such that $\bigcup_{i,j=1,i\neq j}^P {\bar{I}}_{i,j}(t) + B_{2\widetilde r_c}(0)$
is compactly supported in $B_R(0)$ for all times $t\in [0,T_{\mathrm{strong}})$.

Then, for every phase $i\in\{1,\ldots,P\}$ there exist weight functions
\begin{align*}
\vartheta_i\in L^\infty([0,T_{\mathrm{strong}});W^{1,\infty}(\Rd[2];[-1,1]))\cap
W^{1,\infty}([0,T_{\mathrm{strong}});L^\infty(\Rd[2];[-1,1]))
\end{align*}
with $\supp\vartheta_i\subset B_{2R}(0)$ subject to the following additional conditions:
\begin{itemize}[leftmargin=0.7cm]
\item[i)] It holds $\vartheta_i < 0$ in $(\{\bar\chi_i=1\}\cap B_{2R}(0))\setminus\partial\{\bar\chi_i=1\}$
					as well as $\vartheta_i > 0$ in $(\{\bar\chi_i=0\}\cap B_{2R}(0))\setminus\partial\{\bar\chi_i=1\}$.
\item[ii)] It holds $\vartheta_i\equiv 0$ on $\partial\{\bar\chi_i=1\}$.
\item[iii)] We have the bound $|\dist(\cdot,\partial\{\bar\chi_i=1\})|\leq C|\vartheta_i|$
						on $\partial\{\bar\chi_i=1\}{+}B_{2\widetilde r_c}(0)$.
\end{itemize}
Moreover, let the global velocity field $B$ be defined as in Construction~\ref{DefinitionGlobalB}.
Then, we in addition have the following estimate on the advective derivative
\begin{align}\label{AdvectionEquationVolumeControl}
|\partial_t\vartheta_i + (B\cdot\nabla)\vartheta_i| \leq C|\vartheta_i|
\end{align}
for all $i\in\{1,\ldots,P\}$.
\end{lemma}

\begin{proof}
Let $\vartheta\colon\Rd[]\to\Rd[]$ be a
truncation of the identity with $\vartheta(r)=r$ for $|r|\leq\frac{1}{2}$,
$\vartheta(r)= -1$ for $r\leq -1$, $\vartheta(r)=1$ for $r\geq 1$,
$0\leq\vartheta'\leq 2$ and $|\vartheta''|\leq C$.
Let $\eta_{R}\in C^\infty_{\mathrm{cpt}}(\Rd[2];[0,1])$ be any smooth cutoff such that
$\eta_{R}\equiv 1$ on $B_{R}$ and $\supp\eta_R \subset B_{2R}$.
The idea for what follows is to construct in a first step a family of
weight functions $\hat\vartheta_i$ which satisfy all the requirements of
Lemma~\ref{LemmaWeightedVolumeControl} except for $\supp\hat\vartheta\subset B_{2R}(0)$.
Once we succeeded with that we may define $\vartheta_i:=\eta_R\hat\vartheta_i$
in a second step.

\textit{Step 1: Construction of $\hat\vartheta_i$.}
If $\partial\{\bar\chi_i{=}1\}$ consists of only one interface, we then simply define
\begin{align}\label{DefinitionWeightNoTripleJunction}
\hat\vartheta_i(x,t) := \vartheta\Big(\frac{\sigdist(x,\partial\{\bar\chi_i(\cdot,t){=}1\})}{\widetilde r_c}\Big),
\quad (x,t)\in \Rd[2]\times [0,T_{\mathrm{strong}}).
\end{align}

If $\partial\{\bar\chi_i{=}1\}$ consists of more than one two-phase interface, we
first define the weight function $\hat\vartheta_{i}$ away from the triple junctions
by the same formula:
\begin{align}\label{DefinitionWeightTwoPhase}
\hat\vartheta_i(x,t) := \vartheta\Big(\frac{\sigdist(x,\partial\{\bar\chi_i(\cdot,t){=}1\})}{\widetilde r_c}\Big),
\quad t\in[0,T_{\mathrm{strong}}),\,x\in \Rd[2]\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}B_{2\widetilde r_c}(\mathcal{T}_k(t)).
\end{align}

Now, let $k\in\mathcal{K}^{\mathrm{3j}}$ be a triple junction $\mathcal{T}_k\subset\partial\{\bar\chi_i{=}1\}$
with the other two majority phases being $j,p\in\{1,\ldots,P\}$.
As outlined in Definition~\ref{def:triod}, $B_{2\widetilde r_c}(\mathcal{T}_k)$
decomposes into six wedges. The three wedges $W_{i,j}$, $W_{j,p}$ resp.\ $W_{p,i}$,
contain the interfaces ${\bar{I}}_{i,j}$, ${\bar{I}}_{j,p}$ resp.\ ${\bar{I}}_{p,i}$.
The other three are interpolation wedges denoted by $W_{i}$, $W_{j}$ resp.\ $W_{p}$.
Moreover, for the interpolation wedge $W_i$, say, we denote
by $\lambda_{i}^{j}$ the interpolation function as built in Lemma~\ref{lemma:interpolation_functions}
and which is equal to one on $(\partial W_{i,j}\cap\partial W_{i})\setminus\mathcal{T}_{k}$ and
vanishes on $(\partial W_{p,i}\cap\partial W_{i})\setminus\mathcal{T}_{k}$. We also define
$\lambda_{i}^{p}:= 1-\lambda_{i}^{j}$. Analogously, one introduces the interpolation
functions on $W_{j}$ and $W_{p}$.

We now define the weight function $\hat\vartheta_i$ on the ball $B_{2\widetilde r_c}(\mathcal{T}_k)$
as follows:
\begin{align}\label{DefinitionWeightWedgeInterface}
\hat\vartheta_i(x,t) := \vartheta\Big(\frac{\sigdist(x,{\bar{I}}_{i,j}(t))}{\widetilde r_c}\Big),
\quad t\in[0,T_{\mathrm{strong}}),\,x\in W_{i,j}(t)\cap B_{2\widetilde r_c}(\mathcal{T}_k(t)),
\end{align}
and analogously on $W_{i,p}$, whereas we interpolate for $t\in [0,T_{\mathrm{strong}})$ and
$x\in W_{i}(t)\cap B_{2\widetilde r_c}(\mathcal{T}_k(t))$ by means of
\begin{align}\label{DefinitionWeightInterpolationWedge}
\hat\vartheta_i(x,t) :=
\lambda_{i}^{j}(x,t)\vartheta\Big(\frac{\sigdist(x,{\bar{I}}_{i,j}(t))}{\widetilde r_c}\Big)
+\lambda_{i}^{p}(x,t)\vartheta\Big(\frac{\sigdist(x,{\bar{I}}_{i,p}(t))}{\widetilde r_c}\Big).
\end{align}
Furthermore, we define
\begin{align}\label{DefinitionWeightComplementWedge}
\hat\vartheta_i(x,t) := \vartheta\Big(\frac{\dist(x,\mathcal{T}_k(t)}{\widetilde r_c}\Big),
\quad t\in[0,T_{\mathrm{strong}}),\,x\in W_{j,p}(t) \cap B_{2\widetilde r_c}(\mathcal{T}_k(t)),
\end{align}
whereas we again interpolate for $t\in [0,T_{\mathrm{strong}})$ and $x\in W_{j}(t)\cap B_{2\widetilde r_c}(\mathcal{T}_k(t))$ via
\begin{align}\label{DefinitionWeightInterpolationWedge2}
\hat\vartheta_i(x,t) :=
\lambda_{j}^{p}(x,t)\vartheta\Big(\frac{|x-p_k(t)|}{\widetilde r_c}\Big)
+\lambda_{j}^{i}(x,t)\vartheta\Big(\frac{\sigdist(x,{\bar{I}}_{i,j}(t))}{\widetilde r_c}\Big),
\end{align}
and analogously for $t\in [0,T_{\mathrm{strong}})$ and $x\in W_{p}(t)\cap B_{2\widetilde r_c}(\mathcal{T}_k(t))$.

First of all, it is immediate from the definitions that properties \textit{i)}, \textit{ii)}
and \textit{iii)} hold true as required. Furthermore, the asserted regularity
for the weight functions $\hat\vartheta_i$ is a consequence of the regularity of the signed distance
to the interfaces as well as the controlled blowup \eqref{boundslambda1} and \eqref{boundslambda2}
of the derivatives of the interpolation parameter.

For \eqref{AdvectionEquationVolumeControl}, we may restrict ourselves
to the tubular neighborhood $\partial\{\bar\chi_i{=}1\}{+}B_{2\widetilde r_c}(0)$.
By the choice of the localization parameter $\widetilde r_c>0$ from Lemma~\ref{LemmaPartitionOfUnity},
we have that for all $x\in(\partial\{\bar\chi_i{=}1\}{+}B_{2\widetilde r_c}(0))
\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}B_{2\widetilde r_c}(\mathcal{T}_k)$ there is exactly one $m\in\mathcal{K}^{\mathrm{2ph}}$
such that $\sigdist(\cdot,\partial\{\bar\chi_i{=}1\})=\sigdist(\cdot,\mathcal{T}_m)$ in a small
neighborhood around that point. In other words, because of the localization
properties \eqref{LocalizationTwoPhase}--\eqref{LocalizationTwoInterfaces}
for the partition of unity constructed in Lemma~\ref{LemmaPartitionOfUnity},
we have in a neighborhood of such points $B(x)=\eta_m(x)B^m(x)$ by the definition
\eqref{GlobalVelocityField} of $B$.
It therefore follow from \eqref{DefinitionWeightTwoPhase}, \eqref{DefinitionVelocityTwoPhase} and
\eqref{TransportSignedDistanceTwoPhase} that
\begin{align*}
\partial_t\hat\vartheta_i(x) = -(B^m(x)\cdot\nabla)\hat\vartheta_i(x)
=-(B(x)\cdot\nabla)\hat\vartheta_i(x)-(1-\eta_m)(B^m(x)\cdot\nabla)\hat\vartheta_i(x)
\end{align*}
on $(\partial\{\bar\chi_i{=}1\}{+}B_{2\widetilde r_c}(0))
\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}B_{2\widetilde r_c}(\mathcal{T}_k)$.
Since we have the estimate $|1-\eta_m(x)|\leq C|\hat\vartheta_i(x)|$ on
$(\partial\{\bar\chi_i{=}1\}{+}B_{2\widetilde r_c}(0))
\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}B_{2\widetilde r_c}(\mathcal{T}_k)$
by definition \eqref{DefinitionEtaTwoPhaseAwayTripleJunction}
and property \textit{iii)} of $\hat\vartheta_i$, we indeed have \eqref{AdvectionEquationVolumeControl}
for points $x\in(\partial\{\bar\chi_i{=}1\}{+}B_{2\widetilde r_c}(0))
\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}B_{2\widetilde r_c}(\mathcal{T}_k)$ away from the triple junctions.

We move on with checking \eqref{AdvectionEquationVolumeControl} on $B_{2\widetilde r_c}(\mathcal{T}_k)$.
On the wedge $W_{j,p}$, we make use of the evolution equation
$\frac{\mathrm{d}}{\mathrm{d}t}p_k(t)=B(p_k(t),t)$ (where $\mathcal{T}_k(t)=\{p_k(t)\}$) to obtain
\begin{align*}
\partial_t\hat\vartheta_i = -(B(p_k(t),t)\cdot\nabla)\hat\vartheta_i)
= -(B\cdot\nabla)\hat\vartheta_i-\big(\big(B(p_k(t),t){-}B\big)\cdot\nabla\big)\hat\vartheta_i,
\end{align*}
from which \eqref{AdvectionEquationVolumeControl} follows on $W_{i,j}\cap B_{2\widetilde r_c}(\mathcal{T}_k)$ due to the
Lipschitz continuity of the velocity field $B$ and property \textit{iii)} of $\hat\vartheta_i$.
On the wedge $W_{i,j}$, we may compute using \eqref{TransportSignedDistanceTwoPhase}
and \eqref{DefinitionWeightWedgeInterface} (as well as assuming for notational convenience
that $\mathcal{T}_m\subset {\bar{I}}_{i,j}$)
\begin{align*}
\partial_t\hat\vartheta_i &= -(B^m\cdot\nabla)\hat\vartheta_i
\\&
=-(B\cdot\nabla)\hat\vartheta_i
-(1{-}\eta_m{-}\eta_k)(B^m\cdot\nabla)\hat\vartheta_i
-\eta_k\big((B^m{-}B^k)\cdot\nabla\big)\hat\vartheta_i.
\end{align*}
Since $|1-\eta_m-\eta_k|\leq C|\dist(\cdot,{\bar{I}}_{i,j})|^2$ on $W_{i,j}\cap B_{2\widetilde r_c}(\mathcal{T}_k)$,
and also \eqref{ConsistentTangentialComponent} holds true, the bound
\eqref{AdvectionEquationVolumeControl} on $W_{i,j}\cap B_{2\widetilde r_c}(\mathcal{T}_k)$
thus again follows from property \textit{iii)} of $\hat\vartheta_i$.
The validity of \eqref{AdvectionEquationVolumeControl} on $W_{i,p}\cap B_{2\widetilde r_c}(\mathcal{T}_k)$
follows analogously.

It remains to discuss the interpolation wedges. To this end, we only give
the argument for $W_{i}$. The corresponding bound on the other two
interpolation wedges $W_{j}$ resp.\ $W_{p}$ follows similarly. If the
advective derivative drops on the interpolation parameter, we may
rely on \eqref{advectionlambda}, $\vartheta(0)=0$ as well as
property \textit{iii)} of $\hat\vartheta_i$ to obtain on $W_i\cap  B_{2\widetilde r_c}(\mathcal{T}_k)$
the identity
\begin{align*}
&\vartheta\Big(\frac{\sigdist(\cdot,{\bar{I}}_{i,j})}{\widetilde r_c}\Big)\partial_t\lambda_{i}^{j}
+\vartheta\Big(\frac{\sigdist(\cdot,{\bar{I}}_{i,p})}{\widetilde r_c}\Big)\partial_t\lambda_{i}^{p}
\\&
=-\vartheta\Big(\frac{\sigdist(\cdot,{\bar{I}}_{i,j})}{\widetilde r_c}\Big)(B\cdot\nabla)\lambda_{i}^{j}
-\vartheta\Big(\frac{\sigdist(\cdot,{\bar{I}}_{i,j})}{\widetilde r_c}\Big)(B\cdot\nabla)\lambda_{i}^{p}
+ O(\hat\vartheta_i).
\end{align*}
Furthermore, denoting again $\mathcal{T}_m\subset {\bar{I}}_{i,j}$ and introducing
for notational convenience the shorthand
$\vartheta^m_i:=\vartheta\big(\frac{\sigdist(\cdot,{\bar{I}}_{i,j})}{\widetilde r_c}\big)$,
we may compute due to \eqref{TransportSignedDistanceTwoPhase} on the interpolation
wedge $W_{i}\cap  B_{2\widetilde r_c}(\mathcal{T}_k)$
\begin{align*}
\partial_t\vartheta^m_i &= -(B^m\cdot\nabla)\vartheta^m_i
\\&
=-(B\cdot\nabla)\hat\vartheta_i
-\big((B^m{-}B^k)\cdot\nabla\big)\hat\vartheta_i
-\big((B^k{-}B)\cdot\nabla\big)\hat\vartheta_i.
\end{align*}
The latter two terms are again of required order. Since an analogous computation can be carried out for
$\vartheta\big(\frac{\sigdist(\cdot,{\bar{I}}_{i,p})}{\widetilde r_c}\big)$,
we infer that \eqref{AdvectionEquationVolumeControl} indeed holds true
on the interpolation wedge $W_{i}\cap  B_{2\widetilde r_c}(\mathcal{T}_k)$.

\textit{Step 2: Construction and properties of $\vartheta_i$.}
As already mentioned at the beginning of the proof, we may now define
$\vartheta_i:=\eta_R\hat\vartheta_i$. The regularity, the support property
as well as the requirements \textit{i)}--\textit{iii)} for $\vartheta_i$ are then immediate
consequences of its definition and the previous step. The estimate \eqref{AdvectionEquationVolumeControl}
on the advective derivative also carries over since $\eta_R$ is time-independent
and $(B\cdot\nabla)\eta_R\equiv 0$ holds true. The latter is a consequence of $B$
being supported in $\bigcup_{i,j=1,i\neq j}^P {\bar{I}}_{i,j}(t) + B_{2\widetilde r_c}(0)\subset B_R(0)$
(which follows from the definition \eqref{GlobalVelocityField} and the localization properties of the partition
of unity from Lemma~\ref{LemmaPartitionOfUnity}) and that $\supp\nabla\eta_R\subset B_{2R}(0)\setminus B_R(0)$.
This concludes the proof of Lemma~\ref{LemmaWeightedVolumeControl}.
\end{proof}

We have now everything in place to post-process the right-hand side in \eqref{RelativeEntropyInequalityVolumeControl}.

\begin{lemma}
\label{lem:postProcessedRelEntropyInequalityVolume}
Let $d=2$, $P\geq 2$ be an integer and $\sigma\in\Rd[P\times P]$ be an admissible matrix of surface tensions
in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions}.
Let $\chi=(\chi_1,\ldots,\chi_P)$ be a BV solution of multiphase mean curvature flow in
the sense of Definition~\ref{DefinitionBVSolution} on some time interval $[0,T_{\BV})$, and let
$\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution of multiphase mean curvature flow in
the sense of Definition~\ref{DefinitionStrongSolution} on some time interval $[0,T_{\mathrm{strong}})$ with $T_{\mathrm{strong}}\leqT_{\BV}$.

Let $(\xi_{i,j},B)$ be the gradient flow calibration provided by Construction~\ref{DefinitionGlobalXi}
and Construction~\ref{DefinitionGlobalB}. Consider moreover the family of weight functions
$\vartheta_i$ as constructed by Lemma~\ref{LemmaWeightedVolumeControl}. Finally, recall the
definition of the interface error control $E[\chi|\xi]$ from \eqref{DefinitionRelativeEntropyFunctional}
and the definition of the weighted volume error control
$E_{\mathrm{volume}}[\chi|\bar\chi]$ from \eqref{volumeErrorFunctional}.
Then, uniformly over all $\delta\in (0,1)$, it holds
\begin{align*}
E_{\mathrm{volume}}[\chi|\bar\chi](T) &\leq
E_{\mathrm{volume}}[\chi|\bar\chi](0) +
\frac{C}{\delta}\int_0^TE_{\mathrm{volume}}[\chi|\bar\chi](t) + E[\chi|\xi](t) \,\mathrm{d}t
\\&~~~
+\delta\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}(V_{i,j}{-}B\cdot\xi_{i,j})^2\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t
\end{align*}
for almost every $T\in [0,T_{\mathrm{strong}})$.
\end{lemma}

\begin{proof}
Starting point is \eqref{RelativeEntropyInequalityVolumeControl}
meaning that we need to estimate the term $R_{\mathrm{volume}}$.
Note that Lemma~\ref{relativeEntropyVolume} is indeed applicable
with $(\xi_{i,j},B)$ being the gradient flow calibration provided by Construction~\ref{DefinitionGlobalXi}
and Construction~\ref{DefinitionGlobalB}, and $\vartheta_i$ being the family of weight functions
as constructed by Lemma~\ref{LemmaWeightedVolumeControl}.

First, we may infer because of \eqref{AdvectionEquationVolumeControl},
H\"older's and Young's inequality as well as the bound \eqref{eq:L2excess} that
\begin{align*}
|R_{\mathrm{volume}}|
&\leq
C\delta^{-1}\sum_{i,j=1,i\neq j}^{P}
\int_0^T\int_{I_{i,j}(t)\cap\{|\dist(\cdot,\partial\{\bar\chi_i(\cdot,t)=1\})|\leq 2\widetilde r_c\}}
\vartheta_i^2\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t
\\&~~~
+C\delta^{-1}\sum_{i,j=1,i\neq j}^{P}
\int_0^T\int_{I_{i,j}(t)\cap\{|\dist(\cdot,\partial\{\bar\chi_i(\cdot,t)=1\})|> 2\widetilde r_c\}}
\vartheta_i^2\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t
\\&~~~~
+\delta\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}(V_{i,j}{-}B\cdot\xi_{i,j})^2\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t
\\&~~~~
+C\delta^{-1}\int_0^T E[\chi|\xi](t) +
E_{\mathrm{volume}}[\chi|\bar\chi](t)\,\mathrm{d}t
\end{align*}
holds true, uniformly over all $\delta\in (0,1)$. Since $\vartheta_i\equiv 0$
on $\partial\{\bar\chi_i=1\}$ by Lemma~\ref{LemmaWeightedVolumeControl},
we may estimate $|\vartheta_i|^2\leq C|\dist(\cdot,\partial\{\bar\chi_i=1\})|^2$
on $\{|\dist(\cdot,\partial\{\bar\chi_i=1\})|\leq 2\widetilde r_c\}$.
To discuss the contribution on $\{|\dist(\cdot,\partial\{\bar\chi_i(\cdot,t)=1\})|> 2\widetilde r_c\}=:A_i$,
recall first that we denoted by ${\bar{I}}(t)$ the union of all interfaces in the strong
solution. Note that on $A_i\cap\{\dist(\cdot,{\bar{I}})>2\widetilde r_c\}$
we have $\xi_{i,j}\equiv 0$ for all $j\neq i$ by \eqref{LocalizationTwoPhase},
\eqref{LocalizationTripleJunction} and the definition \eqref{GlobalXi}.
On $A_i\cap\{\dist(\cdot,{\bar{I}})\leq 2\widetilde r_c\}$ we at least have $|\xi_{i,j}|\leq c <1$
due to \eqref{CoercivityMinorityPhase} and definition \eqref{GlobalXi}. In summary,
we may estimate $|\vartheta_i|^2\leq C(1-|\xi_{i,j}|)$
on $\{|\dist(\cdot,\partial\{\bar\chi_i(\cdot,t)=1\})|> 2\widetilde r_c\}$.

It is then a consequence of \eqref{eq:coercivityLengthXi} that we may estimate
\begin{align}\label{BoundWeightVol}
|R_{\mathrm{weightVol}}|
&\leq
\delta\sum_{i,j=1,i\neq j}^{P}\sigma_{i,j}\int_0^T\int_{I_{i,j}(t)}
\frac{1}{2}(V_{i,j}{-}B\cdot\xi_{i,j})^2\,\mathrm{d}\mathcal{H}^1\,\mathrm{d}t
\\&~~~~\nonumber
+C\delta^{-1}\int_0^T E[\chi|\xi](t) +
E_{\mathrm{volume}}[\chi|\bar\chi](t) \,\mathrm{d}t
\end{align}
uniformly over all $\delta\in (0,1)$. This concludes the proof.
\end{proof}

We are finally in position to provide a proof for the stability estimate \eqref{StabilityEstimateVolume}.

\begin{proof}[Proof of Proposition~\ref{PropositionVolumeError}]
As a combination of Lemma~\ref{lem:postProcessedRelEntropyInequalityVolume},
the inequality \eqref{eq:postProcessedInterfaceError} and an absorption argument,
we obtain for the sum of the two functionals $E_{\mathrm{volume}}[\chi|\bar\chi]$
resp.\ $E[\chi|\xi]$ defined in \eqref{volumeErrorFunctional} resp.\ \eqref{DefinitionRelativeEntropyFunctional}
the estimate
\begin{align*}
E_{\mathrm{volume}}[\chi|\bar\chi](T) + E[\chi|\xi](T)
\leq C\int_0^T E_{\mathrm{volume}}[\chi|\bar\chi](t) + E[\chi|\xi](t) \,\,\mathrm{d}t
\end{align*}
for almost every $T\in [0,T_{\mathrm{strong}})$. Since the properties of the weight $\vartheta_i$,
see Lemma~\ref{LemmaWeightedVolumeControl}, ensure that $|\vartheta_i(\cdot,t)|$ is comparable to
$\dist(\cdot,\partial\{\bar\chi_i(\cdot,t)=1\})\wedge 1$ this concludes the
proof of Proposition~\ref{PropositionVolumeError} by an application of Gronwall's lemma.
\end{proof}

\section{Gradient flow calibrations at a smooth manifold}
\label{SectionLocalConstructionsTwoPhase}
The aim of this section is to construct a gradient flow calibration
in the simplest situation of one single manifold (with or without boundary) evolving by mean curvature,
see Lemma~\ref{LemmaBoundsLocalConstructionsTwoPhase} for the main result of this section. For the sake of simplicity, we stick to the case $d=2$, but the construction in this section immediately carries over to arbitrary dimensions.
In terms of a gradient flow calibration for a whole network of interfaces in the sense of
Definition~\ref{DefinitionCalibrationGradientFlow}, the vector fields constructed in
Lemma~\ref{LemmaBoundsLocalConstructionsTwoPhase} provide the local building block at a smooth two-phase interface
of the network. These vector fields therefore only live in a small tubular neighborhood around the evolving interface.
In order to make use of the constructions outlined in this section in the general network case,
one has to multiply with a suitably defined cutoff function localizing to the tubular neighborhood
for a given two-phase interface of the network. In light of the coercivity condition in
Definition~\ref{DefinitionCalibrationGradientFlow} of a gradient flow calibration, this cutoff
shall decrease quadratically in the distance to the interface. The construction of such
a set of cutoff functions is deferred to Section~\ref{SectionPartitionOfUnity}.

\begin{definition}\label{DefinitionStrongSolutionTwoPhase}
	Let $T_{\mathrm{strong}}>0$ be a finite time horizon. A one-parameter family $({\bar{I}}(t))_{t\in [0,T_{\mathrm{strong}})}$
	of embedded, connected and orientable one-dimensional $C^4$-manifolds (with or without boundary) in $\Rd[2]$
	is called a \emph{regular interface evolving by mean curvature} if there exists a map
	\begin{align*}
	\Psi\colon {\bar{I}}(0)\times [0,T_{\mathrm{strong}})\to \Rd[2],\quad (x,t)\mapsto \Psi(x,t)=\Psi^t(x)
	\end{align*}
	with the following properties:
	\begin{itemize}
		\item[i)] $\Psi^0=\mathrm{Id}$, and for all $t\in [0,T_{\mathrm{strong}})$ the map
		$\Psi^t\colon {\bar{I}}(0)\to {\bar{I}}(t)$
		is a $C^4$-diffeomorphism such that $\|\Psi\|_{L^\infty_tW^{4,\infty}_x}<\infty$.
		\item[ii)] We have $\partial_t\Psi\in C^0([0,T_{\mathrm{strong}});C^3({\bar{I}}(0);\Rd[2]))$ and
		$\|\partial_t\Psi\|_{L^\infty_tW^{3,\infty}_x}<\infty$.
		\item[iii)] We assume that there
		exists $r_c\in (0,\frac{1}{2}]$ with the following property:
		For all $t\in [0,T_{\mathrm{strong}})$ and all $x\in {\bar{I}}(t)$
		there exists a function $g\colon (-1,1)\to\mathbb{R}$ with $ g'(0)=0$ and $y_{min} \in [-1,0]$
		such that after a translation	and a rotation, ${\bar{I}}(t)\cap B_{2r_c}(x)$ is given by the graph
		$\{(y,g(y))\colon y \in (y_{min},1)\}$, which we require to be a connected set.
		Furthermore, for any of these functions $g$ we require the pointwise bounds
		$|g^{(m)}|\leq r_c^{-(m-1)}$ to hold for all $1\leq m\leq 3$.
	\end{itemize}
	Let $\bar{\vec{n}}(\cdot,t)$ denote a unit normal vector field of ${\bar{I}}(t)$,
	arising by choosing an orientation for ${\bar{I}}(0)$ and transporting it along $\Psi$.
	Let $\bar V(x,t)$ denote the normal speed of a point $x\in {\bar{I}}(t)$, i.e.,
	the normal component of $\partial_t\Psi(y,t)$ at $y=(\Psi^t)^{-1}(x)\in {\bar{I}}(0)$.
	Denoting by $\vec{H}(t)$ the mean curvature vector field of ${\bar{I}}(t)$, we then assume that
	\begin{align}\label{MotionByMeanCurvatureTwoPhase}
	\bar V(x,t)\bar{\vec{n}}(x,t) = \vec{H}(x,t)\quad\text{for all } t\in [0,T_{\mathrm{strong}}),\, x\in {\bar{I}}(t).
	\end{align}
\end{definition}

As usual, tangential velocities can be chosen arbitrarily for a smooth manifold
evolving by its mean curvature. Note also that the preceding definition
allows the boundary points of the manifold, if present, to move.

It follows from our assumptions in Definition~\ref{DefinitionStrongSolutionTwoPhase} that
for all $t\in [0,T_{\mathrm{strong}})$ the maps
\begin{align}\label{DiffeoTubularNeighborhood}
\Phi^t\colon {\bar{I}}(t)\times (-r_c,r_c)\to\Rd[2],\quad
(x,s) \mapsto x + s\bar{\vec{n}}(x,t)
\end{align}
are $C^3$-diffeomorphisms onto their image. Exploiting this property,
we may define a signed distance function in the image of $\Phi^t$ via
the second component of its inverse function
\begin{align}\label{SignedDistanceTwoPhase}
s(x,t) := (\Phi^t)^{-1}_2(x,t)
=
\begin{cases}
					\dist(x,{\bar{I}}(t)), & x\in \Phi^t\big({\bar{I}}(t)\times [0,r_c)\big), \\
					-\dist(x,{\bar{I}}(t)), & x\in \Phi^t\big({\bar{I}}(t)\times (-r_c,0)\big).
					\end{cases}
\end{align}
Let us denote in the following by $\mathcal{U}$ the space-time domain
$\bigcup_{t\in [0,T_{\mathrm{strong}})}\mathrm{im}(\Phi^t)\times\{t\}$.
For a point $(x,t)\in\mathcal{U}$, the projection of $x$ onto the nearest
point on the interface ${\bar{I}}(t)$ is well-defined and given by
\begin{align*}
P_{{\bar{I}}}(x,t) := P_{{\bar{I}}(t)}x := x - s(x,t)\bar{\vec{n}}(x,t).
\end{align*}
As a consequence of our regularity assumptions on the evolving family of interfaces $({\bar{I}}(t))_{t\in [0,T_{\mathrm{strong}})}$,
we infer that the signed distance $s$ (resp.\ its time derivative $\partial_t s$) are of class $C^0_tC^{4}_x$
with $\nabla^4 s \in L^\infty_tL^{\infty}_x$ (resp.\ $C^0_tC^{3}_x$
with $\nabla^3\partial_t s \in L^\infty_tL^{\infty}_x$) in the space-time domain $\mathcal{U}$,
see Lemma~\ref{LemmaPropertiesSignedDistanceTwoPhase}. In particular, the projection map $P_{{\bar{I}}}$
is of class $C^0_t C^{3}_x$ with $\nabla^3 P_{{\bar{I}}} \in L^\infty_tL^\infty_x$ in $\mathcal{U}$.

The identities needed for the estimation of the terms occurring on the right-hand side of
the relative entropy inequality \eqref{RelativeEntropyInequality} in
Proposition~\ref{PropositionRelativeEntropyInequality}, at least on the level
of the two-phase problem considered in this section, are collected in the following result.

\begin{lemma}\label{LemmaBoundsLocalConstructionsTwoPhase}
	Let $({\bar{I}}(t))_{t\in [0,T_{\mathrm{strong}})}$ be a regular interface evolving by mean curvature
	in the sense of Definition~\ref{DefinitionStrongSolutionTwoPhase}. Let
	$\alpha\in L^\infty_tW^{1,\infty}_x$ be an arbitrary map, and define the tangent vector
	field $\bar\tau := J^\mathsf{T} \vec{\bar n}$ where $J$ denotes the counter-clockwise rotation by
	an angle of $90^\circ$.	Then the vector fields $\xi\in L^\infty_tW^{2,\infty}_x\cap W^{1,\infty}_tW^{1,\infty}_x$
	and $B\in L^\infty_tW^{1,\infty}_x$ given by
	\begin{align}
	\label{DefinitionXiTwoPhase}
	\xi(x,t) &:= \nabla s(x,t) = \bar{\vec{n}}(P_{{\bar{I}}(t)}x,t),
	\\
	\label{DefinitionVelocityTwoPhase}
	B(x,t) &:= \vec{H}(P_{{\bar{I}}(t)}x,t) +
	\alpha(P_{{\bar{I}}(t)}x,t)\bar\tau(P_{{\bar{I}}(t)}x,t)
	\end{align}
	for $(x,t)\in  \mathcal{U}:=\bigcup_{t\in [0,T_{\mathrm{strong}})}\mathrm{im}(\Phi^t)\times\{t\}$
	satisfy
	\begin{align}
	\label{TransportEquationXiTwoPhase}
	\partial_t\xi(x,t) + (B(x,t)\cdot\nabla)\xi(x,t)
	+ (\nablaB(x,t))^\mathsf{T}\xi(x,t) &= 0, \\
	\label{LengthConservationTwoPhase}
	\partial_t|\xi(x,t)|^2 + (B(x,t)\cdot\nabla)|\xi(x,t)|^2 &= 0, \\
	\label{DissipTwoPhase}
	B(x,t)\cdot\xi(x,t) + \nabla\cdot\xi(x,t) &= O\big(\dist(x,{\bar{I}}(t))\big)

	\end{align}
	for all $(x,t)\in\mathcal{U}$.
\end{lemma}

\begin{proof}
	It follows from \eqref{TransportSignedDistanceTwoPhase} and  \eqref{SignedDistanceNormal} below,
	as well as from the orthogonality $\bar\tau \cdot \bar{\vec{n}}=0$  that the tangential term in the
	definition of $B$ does not have an effect on the transport equation for
	the signed distance $s$, i.e., we have
	\begin{align*}
	\partial_t s(x,t) = -\big(\vec{H}(P_{{\bar{I}}(t)}x,t)\cdot\nabla\big) s(x,t)
	=-\big(B(x,t)\cdot\nabla\big) s(x,t).
	\end{align*}
	We may take the gradient of this identity so that by definition of $\xi$ we have
	\begin{align*}
	\partial_t\xi(x,t) = \nabla\partial_t s(x,t) =
	-\big(B(x,t)\cdot\nabla\big)\xi(x,t)
	-\big(\nablaB(x,t)\big)^\mathsf{T}\xi(x,t),
	\end{align*}
	which proves \eqref{TransportEquationXiTwoPhase}. The validity of \eqref{LengthConservationTwoPhase}
	is evident from the fact that $|\xi|^2 \equiv 1$.
	For the identity \eqref{DissipTwoPhase}, note first that
	$B(x,t)\cdot\xi(x,t) = \vec{H}(P_{{\bar{I}}(t)}x,t)\cdot\xi(x,t)$
	as a consequence of the orthogonality $\bar\tau \cdot \bar{\vec{n}}=0$. By means of
	the identity \eqref{GradientProjectionInterface} below,
	$$\nabla\bar{\vec{n}}(y,t)|_{y=P_{{\bar{I}}(t)}x}
	=-\big(\vec{H}(P_{{\bar{I}}(t)}x,t)\cdot\bar{\vec{n}}(x,t)\big)
	\bar{\tau}(P_{{\bar{I}}(t)}x,t)\otimes\bar\tau(P_{{\bar{I}}(t)}x,t)$$
	and the definition \eqref{DefinitionXiTwoPhase} of the vector field $\xi$ we compute
	\begin{align*}
	\nabla\cdot\xi(x,t) = -\vec{H}(P_{{\bar{I}}(t)}x,t)\cdot\xi(x,t)
	\big(1{-}s(x,t)\nabla^2s(x,t):\bar{\tau}(P_{{\bar{I}}(t)}x,t)\otimes\bar\tau(P_{{\bar{I}}(t)}x,t)\big).
	\end{align*}
	This concludes the proof.
\end{proof}

The preceding Lemma relies on a number of well-known properties of the signed distance and the nearest point projection.
For further reference, we present them here in a separate statement.

\begin{lemma}\label{LemmaPropertiesSignedDistanceTwoPhase}
Let $({\bar{I}}(t))_{t\in [0,T_{\mathrm{strong}})}$ be a smoothly evolving interface moving by mean curvature
in the sense of Definition~\ref{DefinitionStrongSolutionTwoPhase}. The signed distance function
then satisfies $s\in C^0_tC^{4}_x$ with $\nabla^4 s \in L^\infty_tL^\infty_x$ and $\partial_t s \in C^0_tC^{3}_x$
with $\nabla^3\partial_t s \in L^\infty_tL^\infty_x$ in $\mathcal{U}$.
Its time evolution is given by transport along the flow of the mean curvature vector field:
\begin{align}\label{TransportSignedDistanceTwoPhase}
\partial_t s(x,t) = -\big(\vec{H}(P_{{\bar{I}}(t)}x,t)\cdot\nabla\big) s(x,t),
\quad (x,t)\in\mathcal{U}.
\end{align}
Furthermore, the trivial extensions of the normal and tangent fields
\begin{align*}
\vec{\bar n}(x,t) &:= \vec{\bar n}(P_{{\bar{I}}(t)}x,t),
\\
\bar \tau(x,t) &:= \bar \tau (P_{{\bar{I}}(t)}x,t)= J^\mathsf{T} \vec{\bar n}(P_{{\bar{I}}(t)}x,t),
\end{align*}
where $J$ denotes the counter-clockwise rotation by $90^\circ$,
are transported as well:
\begin{align}\label{TransportNormalVectorTwoPhase}
	\partial_t  \bar{\vec{n}} &= -\big(\vec{H}(P_{{\bar{I}}(t)}x,t)\cdot\nabla\big)\bar{\vec{n}}
	-\big(\nabla\big(\vec{H}(P_{{\bar{I}}(t)}x,t)\big)\big)^\mathsf{T}\bar{\vec{n}},\quad
	(x,t)\in\mathcal{U},
	\\ \label{TransportTangentVectorTwoPhase}
	\partial_t  \bar\tau  &=  -\big(\vec{H}(P_{{\bar{I}}(t)}x,t)\cdot\nabla\big)
	\bar \tau -J^\mathsf{T}\big(\nabla\big(\vec{H}(P_{{\bar{I}}(t)}x,t)\big)\big)^\mathsf{T}J\bar \tau,\quad
	(x,t)\in\mathcal{U}.
\end{align}
The gradient of the projection map is given by
\begin{align}\label{GradientProjectionInterface}
\nabla P_{{\bar{I}}}(x,t) = \mathrm{Id}-\bar{\vec{n}}(P_{{\bar{I}}(t)}x,t)\otimes
\bar{\vec{n}}(P_{{\bar{I}}(t)}x,t) - s(x,t)\nabla^2 s(x,t),
\quad (x,t)\in\mathcal{U}.
\end{align}
Finally, the derivatives of the signed distance $s$ are subject to the relations
\begin{align}
\label{SignedDistanceNormal}
\nabla s(x,t) &= \bar{\vec{n}}(P_{{\bar{I}}(t)}x,t) = \nabla s(y,t)|_{y=P_{{\bar{I}}(t)}x}, \\
\label{LengthConservation1}
\nabla s(x,t)\cdot\partial_t\nabla s(x,t) &= 0, \\
\label{LengthConservation2}
\nabla s(x,t)\cdot\partial_j\nabla s(x,t) &= 0, \quad j=1,\ldots,d, \\
\label{TimeDerivativeSignedDistance}
\partial_t s(x,t) &=\partial_t s(y,t)|_{y=P_{{\bar{I}}(t)}x},
\end{align}
for all points $(x,t)\in\mathcal{U}$.
\end{lemma}

\begin{proof}
The representation of $s$ as a component of the inverse of $\Phi^{t}$ and the first two parts
of Definition~\ref{DefinitionStrongSolutionTwoPhase} initially give the regularity
$s \in C^0_tC^3_x \cap L^\infty_t W^{3,\infty}_x$ and $\partial_t s \in C^0_tC^2_x \cap L^\infty_t W^{2,\infty}_x$.
Apart from identities \eqref{TransportNormalVectorTwoPhase} and \eqref{TransportTangentVectorTwoPhase},
a proof of the remaining, well-known identities was given for instance in \cite[Lemma 10]{FischerHensel} with
the only difference being the precise form of the normal velocity of the evolving family of interfaces.
The equality \eqref{TransportNormalVectorTwoPhase} then follows from the identity \eqref{TransportSignedDistanceTwoPhase}
by differentiation, which in turn implies the equation \eqref{TransportTangentVectorTwoPhase} by rotation.
The higher regularity for the signed distance $s$ and its time derivative $\partial_t s$
finally follows from the identity \eqref{SignedDistanceNormal} and the first two parts
of Definition~\ref{DefinitionStrongSolutionTwoPhase}.
\end{proof}

\section{Gradient flow calibrations at a triple junction}
\label{SectionLocalConstructionsTriod}
The aim of this section is to construct a gradient flow calibration
in the case of three interfaces meeting at a single triple junction.
This geometric setting will be referred to in the following as a \emph{regular triod moving by mean curvature}. All relevant local, geometric properties of a regular triod moving by mean curvature
are collected in Definition~\ref{def:triod}. We then state the main result of this section,
Proposition~\ref{prop:xi_triple_junction}, which provides all relevant properties of the
constructed fields.

The construction of $\xi_{i,j}$ proceeds in three steps.
First, we extend the normal of the interface ${\bar{I}}_{i,j}$ of the strong solution to
auxiliary vector fields $\widetilde\xi_{i,j}$ defined on suitably chosen half-spaces $\mathbb{H}_{i,j}$,
see Figure~\ref{fig:triod}, on which the nearest point-projection onto ${\bar{I}}_{i,j}$ is well-defined and regular.
One should think of $\widetilde\xi_{i,j}$ as the main building block for the vector field
$\xi_{i,j}$ on the half-space $\mathbb{H}_{i,j}$ containing the corresponding interface ${\bar{I}}_{i,j}$.

In the second step, we aim to identify a candidate vector field for the definition of $\xi_{i,j}$
outside the corresponding half-space $\mathbb{H}_{i,j}$. The guiding principle for this step
is to arrange the constructions in such a way that the Herring angle condition at the triple junction
\begin{align}
\label{eq:auxHerring}
\sigma_{1,2}\bar{\vec{n}}_{1,2} + \sigma_{2,3}\bar{\vec{n}}_{2,3} + \sigma_{3,1}\bar{\vec{n}}_{3,1} = 0,
\end{align}
where the indices $1,2,3$ correspond to the phases present at the triple junction,
is satisfied by the family of vector fields $(\xi_{1,2},\xi_{2,3},\xi_{3,1})$
in the whole neighborhood of the triple junction:
\begin{align}
\label{eq:auxHerringAngleConditionExtensions}
	\sigma_{1,2}\xi_{1,2} + \sigma_{2,3} \xi_{2,3} + \sigma_{3,1} \xi_{3,1} = 0.
\end{align}
The motivation for having \eqref{eq:auxHerringAngleConditionExtensions} comes from the observation
that this condition will allow us to define vector fields $(\xi_1,\xi_2,\xi_3)$ such that $\sigma_{i,j}\xi_{i,j}=\xi_i-\xi_j$
holds true. The latter identity in turn is precisely what is needed to compute the time derivative of
the relative entropy functional, see the proof of Proposition~\ref{PropositionRelativeEntropyInequality}
in Section~\ref{SectionDerivationRelativeEntropyInequality}.

How do we achieve \eqref{eq:auxHerringAngleConditionExtensions}? Imagine we want to find candidate
vector fields for $\xi_{2,3}$ and $\xi_{3,1}$ in the half-space $\mathbb{H}_{1,2}$ containing the
interface ${\bar{I}}_{1,2}$ such that \eqref{eq:auxHerringAngleConditionExtensions} is satisfied.
Recall that in the first step we already constructed a candidate vector field $\widetilde\xi_{1,2}$
for $\xi_{1,2}$ in the half space $\mathbb{H}_{1,2}$. However, since the desired condition
\eqref{eq:auxHerringAngleConditionExtensions} really represents an angle condition, the natural
candidates for $\xi_{2,3}$ and $\xi_{3,1}$ in $\mathbb{H}_{1,2}$ are given by appropriate rotations
of $\widetilde\xi_{1,2}$ in order to recover \eqref{eq:auxHerringAngleConditionExtensions};
see Figure~\ref{fig:triple_extension} and Figure~\ref{fig:triple_rotation} above.
The same procedure is repeated on the other two half-spaces $\mathbb{H}_{2,3}$ and $\mathbb{H}_{3,1}$
by appropriately rotating the vector fields $\widetilde\xi_{2,3}$ and $\widetilde\xi_{3,1}$, respectively.

To summarize, we outlined so far the construction of candidate vector fields $\widetilde\xi_{i,j}$ in each of the
half-spaces $\mathbb{H}_{i',j'}$. However, in the regions where the half-spaces overlap (see Figure~\ref{fig:triod})
we now have two competing building blocks for each of the desired vector fields $\xi_{i,j}$.
To overcome this issue, we partition the neighborhood of the triple junction into six wedges
as indicated in Figure~\ref{fig:triod_wedges}, three of which are denoted by $W_{i,j}=W_{j,i}$
and the remaining three by $W_i$. We will require that ${\bar{I}}_{i,j}\subset \overline{W}_{i,j}
\subset\overline{\mathbb{H}}_{i,j}$, see Figure~\ref{fig:triod_wedges}.
The first inclusion corresponds to a geometric smallness condition away from the triple junction.
For the remaining three wedges it is required that $W_i\subset\{\bar\chi_i=1\}$
and $W_i\subset\bigcap_{j\neq i}\mathbb{H}_{i,j}$, see again Figure~\ref{fig:triod_wedges}.
We refer to these wedges from now on as \textit{interpolation wedges}
as they serve as the domains on which we interpolate between the two competing candidate vector fields.

\begin{figure}
	\centering
     \subcaptionbox{\label{fig:triod}}{
	  \centering
 	  \vspace{-\baselineskip}
	  \begin{tikzpicture}[scale=2.9]
	   	\node at (85:.9) {\phantom{$W_{1}$}};
		\node at (185:.9) {\phantom{$W_{2}$}};
		\node at (302:.9) {\phantom{$W_{2}$}};

		\node at (145:.9) {\phantom{$W_{1,2}$}};
		\node at (255:.9) {\phantom{$W_{2,3}$}};
		\node at (17:.9) {\phantom{$W_{3,1}$}};
	  \begin{scope}
	  	\clip (0,0) circle [radius=.8];
	 	\fill[pattern=north west lines,pattern color=blue,opacity=.2] (110:.8) -- (1,1.3) -- (1,-1)--  (290:.95) -- (110:.8);
		\fill[pattern=north east lines,pattern color=red,opacity=.2] (60:.8) -- (-1,1.3) -- (-1,-1)--  (240:.95) -- (60:.8);

	  	\draw[thick,color=red,opacity=.6] (0,0) arc (60:90:2);
		\draw[thick,opacity=.4] (0,0) arc (130:155:2);
		\draw[thick,color=blue] (0,0) arc (110:60:1);

		\draw[color=red,opacity=.6] (60:-.9)--(60:.8);
		\draw[opacity=.4] (130:-.8)--(130:.9);
		\draw[color=blue] (110:-.8)--(110:.8);

		\begin{scope}
			\clip (130:-1.5)--(130:1.5) -- (-2,2) -- (-2,-2) -- (130:-1.5);
		\foreach \i in {-35,...,35}{
				\draw[opacity=.2] (-1,-\i*0.025) -- (1,-\i*0.025);
			};
		\end{scope}

		\end{scope}

			\node[fill=white,inner sep=1pt] at (85:.25){$\bar{\Omega}_1$};
			\node[fill=white,inner sep=1pt] at (190:.35){$\bar{\Omega}_2$};
			\node[fill=white,inner sep=1pt] at (310:.3){$\bar{\Omega}_3$};

			\node at (3:.9){$\bar{I}_{3,1}$};
			\node at (157:.9){$\bar{I}_{1,2}$};
			\node at (230:.9){$\bar{I}_{2,3}$};

	\end{tikzpicture}
     }
     \subcaptionbox{\label{fig:triod_wedges}}{
	  \centering
	 \begin{tikzpicture}[scale=2.9]
	 \begin{scope}
	 	\clip (0,0) circle [radius=0.8];

	 	\begin{scope}
	 		\clip (0,0) -- (70:1.5) --(100:1.5) -- (0,0);
	 		\foreach \i in {0,...,15}
			{
				\draw[color=blue] (-.5,\i*.05) -- (.5,\i*.05);
				\draw[color=red,opacity=.6] (-.5,\i*.05+.025 ) -- (.5,\i*.05 +.025);
			};
	 	\end{scope}

	 		\begin{scope}
	 			\clip (0,0) -- (170:1.5) --(200:1.5) -- (0,0);
	 			\foreach \i in {-15,...,15}
				{
					\draw[opacity=.4] ($(120:1.5) + (210:\i*.05)$) -- ($(120:-1.5) + (210:\i*.05)$);
					\draw[color=red, opacity=.6] ($(120:1.5) + (210:\i*.05+.025)$) -- ($(120:-1.5) + (210:\i*.05+ .025)$);
				};
	 		\end{scope}

	 		\begin{scope}
	 			\clip (0,0) -- (295:1.5) --(305:1.5) -- (0,0);
	 			\foreach \i in {-15,...,15}
				{
					\draw[opacity=.3] ($(240:1.5) + (330:\i*.05)$) -- ($(240:-1.5) + (330:\i*.05)$);
					\draw[color=blue] ($(240:1.5) + (330:\i*.05+.025)$) -- ($(240:-1.5) + (330:\i*.05+ .025)$);
				};
	 		\end{scope}

		\draw[thick,color=red,opacity=.6] (0,0) arc (60:90:2);
		\draw[thick,opacity=.4] (0,0) arc (130:155:2);
		\draw[thick,color=blue] (0,0) arc (110:60:1);

		\draw[dashed,color=red,opacity=.6] (60:-.9)--(60:.8);
		\draw[dashed,opacity=.4] (130:-.8)--(130:.9);
		\draw[dashed,color=blue] (110:-.8)--(110:.8);

		\draw (0,0) -- (70:.8);
		\draw (0,0) -- (100:.8);

		\draw (0,0) -- (170:1);
		\draw (0,0) -- (200:1);

		\draw (0,0) -- (295:.8);
		\draw (0,0) -- (305:.8);

	\end{scope}

		\node at (85:.9) {$W_{1}$};
		\node at (185:.9) {$W_{2}$};
		\node at (302:.9) {$W_{3}$};

		\node at (145:.9) {$W_{1,2}$};
		\node at (255:.9) {$W_{2,3}$};
		\node at (17:.9) {$W_{3,1}$};
	\end{tikzpicture}
     }
     \caption{a) Sketch of a triod with phases $\bar{\Omega}_1$, $\bar{\Omega}_2$, and $\bar{\Omega}_3$; and the corresponding interfaces. The bottom left to top right hatched region is the halfspace $\mathbb{H}_{1,2}$, the horizontally hatched region is $\mathbb{H}_{2,3}$, and the top left to bottom right hatching represents $\mathbb{H}_{3,1}$. b) The interpolation wedges, shown as hatched, are given by $W_1$, $W_2$ and $W_3$. The remaining wedges $W_{1,2}$, $W_{2,3}$ and $W_{3,1}$ contain the corresponding interfaces of the triod. }
        \label{fig:triod_and_wedges}
\end{figure}

The following lemma ensures that we can indeed find wedges with the desired properties.
Its proof is deferred to the end of the section.

\begin{lemma}\label{lem:wedges}
For an admissible matrix of surface tensions $(\sigma_{i,j})_{i,j = 1,2,3}$
in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions},
there exist coefficients $a^{+}_{i}, a^{-}_{i}, b^{+}_{i}, b^{-}_{i} \in \mathbb{R}$ for $i=1,2,3$
such that the following holds: Given a point $p$ and vectors $\tau_{1,2}, \tau_{2,3}, \tau_{3,1} \in \mathbb{S}^1$, which we will below consider to be tangents at the triple junction $p$ (cf.\ Figure~\ref{fig:wedges_proof} below), satisfying
	\begin{align*}
		\sigma_{1,2} \tau_{1,2} +\sigma_{2,3} \tau_{2,3} +\sigma_{3,1} \tau_{3,1} =0
	\end{align*}
	we define the half-spaces
		\begin{align*}

				\mathbb{H}_{i,i+1} & := \left\{x\in\Rd[2]\colon \left(x- p\right)\cdot \tau_{i,i+1} > 0\right\}\\
		\end{align*}
		and the vectors $v_i := a_i^+ \tau_{i,i+1} + a_i^- \tau_{i-1,i}$ and $w_i := b_i^+ \tau_{i,i+1} + b_i^- \tau_{i-1,i}$
		for all $i\in\{1,2,3\}$. Then	these vectors satisfy $v_i,w_i \in \mathbb{S}^1$ and $v_i \cdot w_i \in (0,1)$,
		and the interior of the conical hulls spanned by $(v_i,w_i)$ and $(v_i,w_{i+1})$ define pairwise disjoint,
		non-empty, open wedges
		\begin{align}
		  \label{def:interpolationWedge}
			W_i &:=
			\{ \gamma_1 v_i + \gamma_2 w_i : \gamma_1,\gamma_2 \in (0,\infty)\}
			\\
			\label{def:wedgeInterface}
			W_{i,i+1} & :=
				\{ \gamma_1 v_i + \gamma_2 w_{i+1} : \gamma_1,\gamma_2 \in (0,\infty)\}
		\end{align}
		satisfying
		\begin{align}
		  \label{eq:partitionByWedges}
			\bigcup_{i=1,2,3}(W_i\cup W_{i,i+1}) &= \Rd[2],
			\\
		  \label{eq:condInterpolationWedge}
			W_i & \subset \mathbb{H}_{i,i+1} \cap \mathbb{H}_{i-1,i},
			\\
			\label{eq:condInterfaceWedge}
			W_{i,i+1} & \subset \mathbb{H}_{i,i+1},
			\\
			\label{eq:choiceOrientation}
			\tau_{i,i+1} & \in W_{i,i+1}.
		\end{align}
		In order to not rely on cyclical notation in later sections, we set $W_{i+1,i}:=W_{i,i+1}$
		for all $i\in\{1,2,3\}$.
\end{lemma}

We have everything in place to sketch the third and final step in the construction of
the vector fields $\xi_{i,j}$. To minimize confusion with the indices, let us concentrate
for the sake of the discussion on just one of these vector fields, say $\xi_{1,2}$.
On each of the wedges $W_{i,j}$ we define $\xi_{1,2}$ by means of the first two steps.
I.e., we define $\xi_{1,2}:=\widetilde\xi_{1,2}$ on $W_{1,2}$ and $\xi_{1,2}:=R\widetilde\xi_{2,3}$
(resp.\ $\xi_{1,2}:=R'\widetilde\xi_{3,1}$) on $W_{2,3}$ (resp.\ on $W_{3,1}$) for an appropriate
rotation matrix $R$ (resp.\ $R'$) in order to satisfy \eqref{eq:auxHerringAngleConditionExtensions}.
Note that these definitions are justified because of \eqref{eq:condInterfaceWedge}.

On the interpolation wedges we proceed by gluing the two competing constructions. For example,
on $W_1$ we define $\xi_{1,2}:=(1{-}\lambda_1)\widetilde\xi_{1,2} + \lambda_1R'\widetilde\xi_{3,1}$
whereas on $W_2$ (resp.\ $W_3$) we set $\xi_{1,2}:=(1{-}\lambda_2)R\widetilde\xi_{2,3} + \lambda_2\widetilde\xi_{1,2}$
(resp.\ $\xi_{1,2}:=(1{-}\lambda_3)R'\widetilde\xi_{3,1}+\lambda_3R\widetilde\xi_{2,3}$).
The functions $(\lambda_1,\lambda_2,\lambda_3)$ are suitably chosen interpolation functions,
see Lemma~\ref{lemma:interpolation_functions}. Because of \eqref{eq:condInterpolationWedge}
these definitions are again justified.

As a final remark concerning the construction of the vector fields $\xi_{i,j}$, it is not clear
at this point whether it is possible to carry out the above program together with having
sufficiently high regularity for the vector fields $\xi_{i,j}$ at the triple junction.
The naive candidate for the auxiliary vector fields $\widetilde\xi_{i,j}$ from the first step
of the construction are the gradient flow calibrations at a smooth manifold from the previous section, i.e.,
$\widetilde\xi_{i,j}(x):=\bar{\vec{n}}_{i,j}(P_{{\bar{I}}_{i,j}}(x))$
on $\mathbb{H}_{i,j}$ where $P_{{\bar{I}}_{i,j}}$
denotes the nearest point projection
onto ${\bar{I}}_{i,j}$. However, this ansatz together with the other two steps only provides
continuous vector fields $\xi_{i,j}$ which already fail to be Lipschitz at the triple junction.
Hence, we have to employ in the first step of the construction a careful adaption of the
naive candidate for the auxiliary vector fields $\widetilde\xi_{i,j}$, see \eqref{AnsatzAuxiliaryXiHalfSpace}.

Following the work in \cite{MantegazzaNovagaPludaSchulze}, we now define a notion of a regular triod moving
by mean curvature. In addition to the framework outlined in \cite{MantegazzaNovagaPludaSchulze},
we further impose certain ``smallness'' conditions on the geometry. In the case of a general network moving by
its mean curvature, these conditions will be satisfied after localizing around the triple junctions
on a sufficiently small scale, see Lemma~\ref{LemmaPartitionOfUnity}.

\begin{definition}
\label{def:triod}
Let $r>0$ be a radius and $T_{\mathrm{strong}}>0$ be a finite time horizon.
Let $p\colon [0,T_{\mathrm{strong}})\to\Rd[2]$ be a $C^1$-map.
We call a measurable map
\begin{align*}
\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)\colon
\bigcup_{t\in [0,T_{\mathrm{strong}})}B_r(p(t))\times\{t\}\to\{0,1\}^3
\end{align*}
a \emph{regular triod with triple junction $p(t)$ evolving by mean curvature
in the space-time domain $\bigcup_{t\in [0,T_{\mathrm{strong}})}B_{r}(p(t))\times\{t\}$},
if it satisfies the following requirements:
	\begin{itemize}[leftmargin=0.7cm]
		\item[i)] The map $\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)$ is a strong
							solution for multiphase mean curvature flow in the space-time domain
							$\bigcup_{t\in [0,T_{\mathrm{strong}})}B_{r}(p(t))\times\{t\}$
							in the sense of Definition~\ref{DefinitionStrongSolution}.
		\item[ii)] The phases $\{\bar\chi_i(\cdot,t)=1\} \subset B_r(p(t))$ are open, non-empty
						   and simply connected sets
							 for all $i\in\{1,2,3\}$ and all $t \in [0,T_{\mathrm{strong}})$.
							 Define for all $i,j\in\{1,2,3\}$ the interfaces ${\bar{I}}_{i,j}(t):= \big(\partial\{\bar\chi_i=1\}\cap\partial\{\bar\chi_j=1\}\big)\setminus\partial B_r(p(t))$.
							 Then we assume that the three interfaces intersect only in the single point $p(t)$.
		\item[iii)] We choose the tangent vector $\bar{\tau}_{i,j}(p(t),t) :=
		J^{-1}\bar{\vec{n}}_{i,j}(p(t),t)$ for $i,j \in \{1,2,3\}$ with $i\neq j$, where
			\begin{align}
			\label{eq:rotation}
				J:= \begin{pmatrix}
					0 & -1\\
					1 & 0
				\end{pmatrix}
			\end{align}
			and denote the corresponding wedges constructed in Lemma~\ref{lem:wedges} by $W_{i}(t)$ and $W_{i,i+1}(t)$.
			In this notation, we require ${\bar{I}}_{i,i+1}(t) \subset W_{i,i+1}(t)\cup \{p(t)\}$.
			\item[iv)] We assume that for all $t\in [0,T_{\mathrm{strong}})$ and all $i\in\{1,2,3\}$ the maps
			\begin{align}\label{NormalCoordinatesTriod}
			\Phi^t_{i,i+1}\colon {\bar{I}}_{i,i+1}(t)\times (-2r,2r) \to \Rd[2], \quad
			(x,s) \mapsto x + s\bar{\vec{n}}_{i,i+1}(x,t),
			\end{align}
			are $C^3$-diffeomorphisms onto their image with bounded and continuous third derivatives.
			Moreover we require that the image of	$\Phi^t_{i,i+1}$ contains $\overline{\mathbb{H}}_{i,i+1}(t)\cap B_r(p(t))$.
			Here, the half-space $\mathbb{H}_{i,i+1}(t)$ is as in Lemma~\ref{lem:wedges},
			corresponding to the tangent vector $\bar{\tau}_{i,i+1}(p(t),t)$.
	\end{itemize}
\end{definition}

We are now in a position to state the existence of a gradient flow calibration for a regular triod
moving by mean curvature.

\begin{proposition}
\label{prop:xi_triple_junction}
Let $\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)$ be a regular triod with triple junction $p(t)$
evolving by mean curvature in the sense of Definition~\ref{def:triod} on the space-time domain
$\bigcup_{[0,T_{\mathrm{strong}})} B_{r}(p(t))\times\{t\}$. Denote by ${\bar{I}}(t):=\bigcup_{i\neq j}{\bar{I}}_{i,j}(t)$
the union of the three interfaces. There then exists a radius $\tilde r = \tilde r(\bar\chi) \leq r$,
only depending on the data of $\bar\chi$, with the following property: for all $i,j \in \{1,2,3\}$ with $i\neq j$,
there exist extensions $\xi_{i,j} \in L^\infty_tW^{2,\infty}_x\cap W^{1,\infty}_tC^0_x$
of the unit-normal vector fields, and a velocity field $B \in L^\infty_tW^{1,\infty}_x$,
which are defined on the space-time domain $\bigcup_{[0,T_{\mathrm{strong}})} B_{\tilde r}(p(t))\times\{t\}$
and subject to the following properties:
\begin{itemize}[leftmargin=0.7cm]
		\item[i)] It holds $\xi_{i,j}(x,t) = \bar{\vec{n}}_{i,j}(x,t)$ for all $t\in [0,T_{\mathrm{strong}})$ and
							for all $x \in {\bar{I}}_{i,j}(t)$. We also have $|\xi_{i,j}(x,t)|=1$ for all $t\in [0,T_{\mathrm{strong}})$ and
							for all $x\in B_{\tilde r}(p(t))$.
		\item[ii)] We have the skew-symmetry relation $\xi_{i,j}=-\xi_{j,i}$.
		\item[iii)] The family of vector fields $(\xi_{i,j})_{i\neq j}$ satisfies the Herring angle condition \eqref{eq:auxHerring}
		            in the entire neighborhood of the triple junction, i.e., it holds
			\begin{align}
			\label{xi_compatibility}
				\sigma_{1,2}\xi_{1,2}(x,t) + \sigma_{2,3}\xi_{2,3}(x,t) + \sigma_{3,1}\xi_{3,1}(x,t) = 0
			\end{align}
			for all $t\in [0,T_{\mathrm{strong}})$ and all $x\in B_{\tilde r}(p(t))$.
		\item[iv)] Throughout $\bigcup_{[0,T_{\mathrm{strong}})} B_{\tilde r}(p(t))\times\{t\}$, we have the bounds
			\begin{align}
				\label{TransportEquationXiTriod}
				|\partial_t\xi_{i,j} + (B\cdot\nabla)\xi_{i,j} + (\nablaB)^\mathsf{T}\xi_{i,j}|
				&\leq C\dist(x,{\bar{I}}(t)),
				\\
				\label{DissipTriod}
				|(\nabla\cdot\xi_{i,j}) + B\cdot\xi_{i,j}| &\leq C\dist(x,{\bar{I}}(t)),
				\\
				\label{LengthConservationXiTriodNormalized}
				\xi_{i,j}\cdot\partial_t\xi_{i,j} + \xi_{i,j}\cdot(B\cdot\nabla)\xi_{i,j} &= 0
			\end{align}
			for all $i,j\in\{1,2,3\}$ with $i\neq j$.
			The constant $C>0$ only depends on the regular triod $\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)$,
			and therefore in particular on the parameter $r_c>0$ from Definition~\ref{DefinitionSmoothlyEvolvingPartition}
			of a smoothly evolving network of interfaces.
\item[v)] Finally, the following estimates on the derivatives of $(\xi_{i,j},B)$ hold true
\begin{align}
\label{boundDerivativesXi}
|\nabla^k\xi_{i,j}| &\leq Cr_c^{-k-2}, \quad k\in\{1,2\},
\\\label{boundDerivativeB}
|\nabla B| &\leq Cr_c^{-3}
\end{align}
throughout the space-time domain $\bigcup_{[0,T_{\mathrm{strong}})} B_{\tilde r}(p(t))\times\{t\}$.
\end{itemize}
\end{proposition}
When applying these bounds in Section~\ref{sec:networkConstruction} the radius $\tilde r>0$ will always be fixed and proportional to $r_c$.

\subsection{Local construction close to individual interfaces}\label{subsection:step_1}
Let $\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)$ be a regular triod
with triple junction $p(t)$ moving by mean curvature in the sense of Definition~\ref{def:triod}
on the space-time domain $\bigcup_{[0,T_{\mathrm{strong}})} B_{r}(p(t))\times\{t\}$.
To simplify notation, we will from now on identify indices if they are equivalent mod $3$, i.e.,
we define $\chi_{4}:=\chi_1$, $\chi_5:=\chi_2$, $\chi_0:=\chi_3$, and so on.
In this subsection, we first introduce certain auxiliary vector fields $\widetilde\xi_{i,i+1}$
as extensions of the normal $\bar{\vec{n}}_{i,i+1}$ to the closure of the half-spaces $\mathbb{H}_{i,i+1}$.
Their restriction to the wedge $W_{i,i+1}$ will later serve as the definition of $\xi_{i,i+1}$
on $W_{i,i+1}$. Additionally, they will also serve as building blocks for the interpolation on the wedges $W_i$, see Section~\ref{subsection:step_2}.

We would like to define $\widetilde \xi_{i,i+1}$, and later also the velocity field $B$, by an expansion ansatz in terms of
the signed distance function $s_{i,i+1}$ to the interface ${\bar{I}}_{i,i+1}$.
Making use of the diffeomorphisms from property \textit{iv)} of Definition~\ref{def:triod}
of a regular triod, in analogy to Section \ref{SectionLocalConstructionsTwoPhase} we define
the signed distance function as
\begin{align}
  \nonumber
	s_{i,i+1}(x,t) &:= (\Phi_{i,i+1}^t)^{-1}_2(x,t)
	\\
	& =\begin{cases}
		\mathrm{dist}\big(x,{\bar{I}}_{i,i+1}(t)\big)
		& \text{ for } x\in \Phi^t_{i,i+1}\big({\bar{I}}_{i,i+1}(t)\times [0,2r)\big),\\
		-\mathrm{dist}\big(x,{\bar{I}}_{i,i+1}(t)\big)
		& \text{ for } x\in \Phi^t_{i,i+1}\big({\bar{I}}_{i,i+1}(t)\times (-2r,0)\big)
	\end{cases}
\end{align}
for $x \in \overline{\mathbb{H}}_{i,i+1}(t) \cap B_r(p(t))$. Note that with this choice
we have $\bar{\vec{n}}_{i,i+1}=\nabla s_{i,i+1}$ on the interface ${\bar{I}}_{i,i+1}$.
We extend $\bar{\vec{n}}_{i,i+1}$ to a vector field on $\overline{\mathbb{H}}_{i,i+1}(t) \cap B_r(p(t))$
by means of this relation, and denote this extension still by $\bar{\vec{n}}_{i,i+1}$.
The nearest point projection $P_{i,i+1}$ onto ${\bar{I}}_{i,i+1}$ is given by
\begin{align}
\label{relation_npp_sd}
P_{i,i+1}(x,t) := x - s_{i,i+1}(x,t)\bar{\vec{n}}_{i,i+1}(x,t),
\end{align}
and its gradient by
\begin{align}
\label{gradient_npp}
\nabla P_{i,i+1}(x,t) = \mathrm{Id} - \bar{\vec{n}}_{i,i+1}(x,t)\otimes \bar{\vec{n}}_{i,i+1}(x,t)
-s_{i,i+1}(x,t)\nabla\bar{\vec{n}}_{i,i+1}(x,t).
\end{align}
The ansatz for the extension $\widetilde\xi_{i,i+1}$ of the normal vector field $\bar{\vec{n}}_{i,i+1}|_{{\bar{I}}_{i,i+1}}$ then is
\begin{equation}
\label{AnsatzAuxiliaryXiHalfSpace}
\begin{aligned}
	\widetilde\xi_{i,i+1}(x,t)
	& :=
	\bar{\vec{n}}_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
	\\&~~~~
	+\alpha_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
	s_{i,i+1}(x,t)\bar{\tau}_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
	\\&~~~~
	-\frac{1}{2} \alpha^2_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
	s_{i,i+1}^2(x,t)\bar{\vec{n}}_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
	\\&~~~~
	+\frac{1}{2} \beta_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
	s_{i,i+1}^2(x,t)\bar{\tau}_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
\end{aligned}
\end{equation}
and $\widetilde \xi_{i+1,i} := - \widetilde \xi_{i,i+1}$ for $t\in [0,T_{\mathrm{strong}})$,
$x \in \mathbb{H}_{i,i+1}(t)\cap B_r(p(t))$, and $i\in\{1,2,3\}$.
Here, for every $i\in\{1,2,3\}$ the map
$\alpha_{i,i+1}\colon\bigcup_{[0,T_{\mathrm{strong}})}{\bar{I}}_{i,i+1}(t)\times\{t\} \to \mathbb{R}$
is the solution to the following ODE on the interface ${\bar{I}}_{i,i+1}$ with initial
condition on the triple junction $p(t)$
\begin{align}\label{ODE_alpha}
  \begin{cases}
	\alpha_{i,i+1}(p(t),t) & = \bar{\tau}_{i,i+1}(p(t),t)\cdot\frac{\mathrm{d}}{\mathrm{d}t}p(t) \\
	\left(\bar{\tau}_{i,i+1}(x,t) \cdot \nabla \right) \alpha_{i,i+1}(x,t) & = H_{i,i+1}^2(x,t),
	\qquad\qquad\quad x\in {{I}}_{i,i+1}(t),
  \end{cases}
\end{align}
where $H_{i,i+1}=-\Delta s_{i,i+1}$   is the scalar mean curvature with respect to the normal $\bar{\vec{n}}_{i,i+1}$
on the interface ${\bar{I}}_{i,i+1}$. We extend $H_{i,i+1}$ to a function on
$\overline{\mathbb{H}}_{i,i+1}(t) \cap B_r(p(t))$ by means of
\begin{align}
\label{eq:extensionScalarMeanCurvature}
H_{i,i+1}(x,t) := -\Delta s_{i,i+1}(y,t)|_{y=P_{{\bar{I}}_{i,i+1}(t)}x}.
\end{align}
Note that as a consequence of the mean curvature flow equation \eqref{TransportSignedDistanceTwoPhase} and
the relation \eqref{TimeDerivativeSignedDistance} we have on $\overline{\mathbb{H}}_{i,i+1}(t) \cap B_r(p(t))$
\begin{align}
\label{eq:alternativeReprScalarMeanCurvature}
-H_{i,i+1}(x,t) = \partial_ts_{i,i+1}(y,t)|_{y=P_{{\bar{I}}_{i,i+1}(t)}x} = \partial_t s_{i,i+1}(x,t).
\end{align}
The functions $\beta_{i,i+1}\colon\bigcup_{[0,T_{\mathrm{strong}})}{\bar{I}}_{i,i+1}(t)\times\{t\} \to \mathbb{R}$
are defined as
\begin{align}\label{def_beta}
	\beta_{i,i+1}(x,t) := - \alpha_{i,i+1}(x,t) H_{i,i+1}(x,t)
	- (\bar{\tau}_{i,i+1}(x,t) \cdot \nabla) H_{i,i+1}(x,t).
\end{align}

We briefly present the regularity properties of $\widetilde \xi_{i,i+1}$.

\begin{lemma}
\label{lemma:regularity_tilde_xi}
Let the assumptions and notation of Proposition~\ref{prop:xi_triple_junction} be in place.
For all phases $i\in\{1,2,3\}$, the auxiliary vector field
$\widetilde\xi_{i,i+1}$ is of class $C^0_tC^2_x\cap C^1_tC^0_x$
in the space-time domain $\bigcup_{t\in [0,T_{\mathrm{strong}})}\big(\overline{\mathbb{H}}_{i,i+1}(t)\cap B_r(p(t))\big)\times\{t\}$
with bounded highest order derivatives. More precisely, we have the estimates
\begin{align}
\label{eq:boundsDerivativesTildeXi}
|\nabla^k\widetilde\xi_{i,i+1}| \leq Cr_c^{-k-2},\quad k\in\{0,1,2\}.
\end{align}
Here, $r_c>0$ is the parameter from Definition~\ref{DefinitionSmoothlyEvolvingPartition}
of a smoothly evolving network of interfaces.
\end{lemma}

\begin{proof}
The asserted regularity essentially follows from the regularity of the signed distance function
and its time derivative. In fact, Lemma \ref{LemmaPropertiesSignedDistanceTwoPhase} applies, so that we obtain $s\in C^0_tC^{4}_x \cap C^1_tC^3_x$ with $\nabla^4 s \in L^\infty_tL^\infty_x$
and $\nabla^3\partial_t s \in L^\infty_tL^\infty_x$ in the space-time domain
$\mathcal{H}_{i,i+1}:=\bigcup_{t\in [0,T_{\mathrm{strong}})}\big(\overline{\mathbb{H}}_{i,i+1}(t)\cap B_r(p(t))\big)\times\{t\}$.
Consequently, the projection $P_{i,i+1}$ is of class $C^0_tC^3_x\cap C^1_tC^2_x$ in $\mathcal{H}_{i,i+1}$
due to \eqref{SignedDistanceNormal} and \eqref{TimeDerivativeSignedDistance}.

For what follows, we perform a slight abuse of notation by writing
$\bar{\vec{n}}_{i,i+1}(x,t) = \bar{\vec{n}}_{i,i+1}(P_{i,i+1}(x,t),t),\,
\alpha_{i,i+1}(x,t)=\alpha_{i,i+1}(P_{i,i+1}(x,t),t)$ and so on. It follows
from the previous considerations that $\bar{\vec{n}},\bar{\vec{\tau}}\in C^0_tC^3_x\cap C^1_tC^2_x$.
We immediately infer from the definition \eqref{eq:extensionScalarMeanCurvature},
the regularity of the signed distance function and the regularity of the
nearest point projection that $\partial_t H_{i,i+1}\in C^0_tC^1_x$ as well as $\nabla\partial_t H_{i,i+1}\in C^0_tC^0_x$.
It follows from the alternative representation \eqref{eq:alternativeReprScalarMeanCurvature}
that $H_{i,i+1}\in C^0_tC^3_x$.

In the proof of Lemma~\ref{lemma:compatibility_xi} below,
we will verify that $H_{i,i+1}\vec{\bar{n}}_{i,i+1} + \alpha_{i,i+1}\bar{\tau}_{i,i+1}
=H_{j,j+1}\vec{\bar{n}}_{j,j+1} + \alpha_{j,j+1}\bar{\tau}_{j,j+1}$ holds true
for all $i\neq j$ at the triple junction $p(t)$, see the argument following \eqref{eq:compatiblity_xi}.
Multiplying this identity with the tangent $\bar{\tau}_{i,i+1}(p(t))$
and defining $c_{i,j}(t):=\vec{\bar{n}}_{i,i+1}(p(t))\cdot\vec{\bar{n}}_{j,j+1}(p(t))$ and
$d_{i,j}(t):=\vec{\bar{n}}_{i,i+1}(p(t))\cdot\bar{\tau}_{j,j+1}(p(t))$ (which are constant in time
as they can be expressed in terms of the given matrix of surface tensions) yields
$\alpha_{i,i+1}(p(t))=H_{j,j+1}(p(t))d_{i,j}+\alpha_{j,j+1}(p(t))c_{i,j}$ for all $i\neq j$.
Switching the roles of $i$ and $j$ in the previous formula entails $(1{-}c_{i,j}^2)\alpha_{i,i+1}(p(t))
=H_{j,j+1}(p(t))d_{i,j}+H_{i,i+1}(p(t))d_{i,j}c_{i,j}$ for all $i\neq j$. In particular,
we may express the tangential component $\alpha_{i,i+1}$ at the triple point
solely in terms of the scalar mean curvatures and the given matrix of surface tensions
since $|c_{i,j}|<1$ for all $i\neq j$. The latter condition is a consequence of the fact
that the angle between $\vec{\bar{n}}_{i,i+1}(p(t))$ and $\vec{\bar{n}}_{j,j+1}(p(t))$
is in $(0,\pi)$ as the surface tensions satisfy the triangle inequality.

We can infer from the previous discussion that the initial value of the ODE \eqref{ODE_alpha}
is of class $C^1_t$, whereas the right hand side in the equation for $\alpha_{i,i+1}$
is of class $C^0_tC^3_x\cap C^1_tC^2_x$. Since the nearest point projection enjoys the
same regularity, it follows that $\alpha_{i,i+1}\in C^0_tC^3_x\cap C^1_tC^2_x$.
It is then a direct consequence of the definition \eqref{def_beta}, the regularity
of the nearest point projection as well as the regularity of $\alpha_{i,i+1}$ and $H_{i,i+1}$ that
$\beta_{i,i+1}\in C^0_tC^2_x\cap C^1_tC^0_x$. In particular, the asserted regularity
of the auxiliary vector field $\widetilde\xi_{i,i+1}$ in the space-time domain $\mathcal{H}_{i,i+1}$
now follows immediately from the ansatz \eqref{AnsatzAuxiliaryXiHalfSpace}.

Observe that it follows from Definition~\ref{DefinitionSmoothlyEvolvingPartition}
of a smoothly evolving interface that
\begin{align}
\label{Bound2ndDerivativeSignedDistance}
|\nabla^{k+1}\sigdist(x,I(t))|\leq Cr_c^{-k},\quad k\in\{1,2,3\},
\end{align}
which in particular entails the following bounds for the nearest-point projections
due to \eqref{relation_npp_sd} and the (extensions of the) scalar mean curvatures
due to \eqref{eq:extensionScalarMeanCurvature}
\begin{align}\label{BoundMeanCurvature}
|\nabla^k P_{i,i+1}| \leq Cr_c^{-k},\quad |\nabla^{k} H_{i,i+1}| \leq Cr_c^{-k-1},\quad k\in\{0,1,2\}.
\end{align}
It thus follows from \eqref{ODE_alpha} and \eqref{def_beta} that
\begin{align}\label{BoundCoefficients}
|\nabla^k\alpha_{i,i+1}| \leq Cr_c^{-k},\quad |\nabla^k\beta_{i,i+1}| \leq Cr_c^{-k-2},\quad k\in\{0,1,2\}.
\end{align}
The asserted bounds for the derivatives of the vector fields $\widetilde\xi_{i,i+1}$
can now be inferred from \eqref{AnsatzAuxiliaryXiHalfSpace}. This concludes the proof.
\end{proof}

Ultimately, the point of the ansatz \eqref{AnsatzAuxiliaryXiHalfSpace}
is to ensure \eqref{xi_compatibility} throughout $B_r(p(t))$ together with sufficiently high regularity of $\xi_{i,j}$
at the triple junction. Moreover, the relations \eqref{ODE_alpha} and \eqref{def_beta} also
holding true away from the triple junction turns out to be crucial to obtain the
estimates \eqref{TransportEquationXiTriod} and \eqref{DissipTriod}
on the whole space-time domain. The first step towards these goals are the following relations,
which in particular yield that---after rotation $R_{(i,j)}$---the vector fields
are compatible to second order at the triple junction:

\begin{lemma}
\label{lemma:compatibility_xi}
  Let the assumptions and notation of Proposition~\ref{prop:xi_triple_junction} be in place.
  For each pair $i,j\in\{1,2,3\}$ there exist uniquely determined rotations $R_{(i,j)} \in SO(2)$,
	only depending on the admissible matrix of surface tensions $(\sigma_{i,j})_{i,j = 1,2,3}$
	for the given regular triod $\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)$, with
	\begin{align}\label{def_rotation}
		\bar{\vec{n}}_{i,i+1}(p(t),t) & = R_{(i,j)} \bar{\vec{n}}_{j,j+1}(p(t),t)
	\end{align}
	for all $t\in [0,T_{\mathrm{strong}})$, and
	\begin{align}
		R_{(i,j)} R_{(j,i)}& = \operatorname{Id},\label{rotation_inverse}\\
		R_{(i,i-1)}R_{(i-1,i+1)}R_{(i+1,i)} &= \operatorname{Id}.\label{rotation_full_circle}
	\end{align}
	Furthermore, the ansatz \eqref{AnsatzAuxiliaryXiHalfSpace} satisfies
	the following second-order compatibility conditions at the triple junction:
	\begin{align}
	  \label{compatibility_xi_zero}
		\widetilde \xi_{i,i+1}(p(t),t) &= R_{(i,j)} \widetilde \xi_{j,j+1}(p(t),t),
		\\
		\label{compatibility_xi_first}
		\nabla \widetilde \xi_{i,i+1}(p(t),t) & =
		\nabla \big( R_{(i,j)} \widetilde \xi_{j,j+1} \big)(p(t),t),
		\\
		\label{compatibility_xi_second}
		\nabla^2 \widetilde \xi_{i,i+1}(p(t),t)
		& = \nabla^2 \big( R_{(i,j)} \widetilde \xi_{j,j+1}\big)(p(t),t)
	\end{align}
	for all $t\in [0,T_{\mathrm{strong}})$.
\end{lemma}

\begin{proof}
	It is immediate from the ansatz \eqref{AnsatzAuxiliaryXiHalfSpace} and \eqref{def_rotation} that
	the zero-order condition \eqref{compatibility_xi_zero} is satisfied.
	The two properties \eqref{rotation_inverse} and \eqref{rotation_full_circle} follow from
	\begin{align}
		R_{(i,j)} R_{(j,i)} \bar{\vec{n}}_{i,i+1} & = \bar{\vec{n}}_{i,i+1},\\
		R_{(i,i-1)}R_{(i-1,i+1)}R_{(i+1,i)}\bar{\vec{n}}_{i,i+1} & = \bar{\vec{n}}_{i,i+1},
	\end{align}
	which follow straightforwardly from iterating \eqref{def_rotation}.
	Therefore, it is sufficient to prove the remaining two statements \eqref{compatibility_xi_first}
	and \eqref{compatibility_xi_second} for $j=i+1$ as it then follows automatically for $j=i-1$
	by \eqref{rotation_inverse} and	\eqref{rotation_full_circle} that
	\begin{align*}
	\nabla \big( R_{(i,i-1)} \widetilde \xi_{i-1,i} \big)(p(t),t)
	&= R_{(i,i+1)}\nabla \big( R_{(i+1,i-1)} \widetilde \xi_{i-1,i} \big)(p(t),t)
	\\&
	= R_{(i,i+1)}\nabla \big( \widetilde \xi_{i+1,i-1} \big)(p(t),t)
	\\&
	=  \nabla \big( \widetilde R_{(i,i+1)}\xi_{i+1,i-1} \big)(p(t),t)
	= \nabla\widetilde\xi_{i,i+1}(p(t),t),
	\end{align*}
	and analogously for the second derivative.

	We extend all functions defined on $\bar{I}_{i,i+1}$ to
	$\mathbb{H}_{i,i+1}\cap B_r(p)$ via the projection $P_{i,i+1}$.
	For ease of notation, we also fix the index $i$ and omit all indices, superscripts,
	and arguments for the rest of the proof unless specifially required otherwise.
	The ansatz \eqref{AnsatzAuxiliaryXiHalfSpace} then reads
	\begin{equation}\label{AnsatzXiWedge}
		\widetilde \xi =
		\bar{\vec{n}} + \alpha s \bar{\tau}
		- \frac{1}{2} \alpha^2 s^2 \bar{\vec{n}} +\frac{1}{2}\beta s^2 \bar{\tau}.
	\end{equation}

Using that $\bar{\vec{n}}$ has been extended via $P_{i,i+1}$, as well as the representation
\eqref{relation_npp_sd} and $\bar{\vec{n}} = \nabla s$, we then get
\begin{align}
  \label{eq:derivativeNormal}
	\nabla \bar{\vec{n}} &= - H(1+sH) \bar{\tau} \otimes \bar{\tau} +O(s^2),
	\\
	\label{eq:derivativeTangent}
	\nabla \bar{\tau} &= \phantom{+} H(1+sH) \bar{\vec{n}}\otimes\bar{\tau}+O(s^2),
\end{align}
As a result we infer from the choice \eqref{ODE_alpha} that
\begin{equation}\label{eq:xi_first_derivative}
\begin{aligned}
		\nabla \widetilde \xi & =  - H(1 + sH)\, \bar{\tau} \otimes \bar{\tau}
		+ s(\bar\tau\cdot\nabla)\alpha \, \bar{\tau} \otimes \bar\tau
		+ (\alpha + \beta s) \, \bar{\tau} \otimes \bar{\vec{n}}
		\\
		& \quad + \alpha  H s \, \bar{\vec{n}} \otimes \bar{\tau}
		- \alpha^2 s \, \bar{\vec{n}} \otimes \bar{\vec{n}} + f
		\\&
		\stackrel{\eqref{ODE_alpha}}{=}  - H\, \bar{\tau} \otimes \bar{\tau}
		+ (\alpha + \beta s) \, \bar{\tau} \otimes \bar{\vec{n}}
		  + \alpha  H s \, \bar{\vec{n}} \otimes \bar{\tau}
		- \alpha^2 s \, \bar{\vec{n}} \otimes \bar{\vec{n}} + f
\end{aligned}
\end{equation}
where $f\in O(s^2)$ such that $\nabla f\in O(|s|)$. At the triple junction this gives
\begin{align}\label{eq:xi_first_derivative_tj}
		\nabla \widetilde \xi(p) & = - H \, \bar{\tau} \otimes \bar{\tau}  + \alpha \, \bar{\tau} \otimes \bar{\vec{n}}.
\end{align}
Carefully noting that $\alpha (\nabla \bar{\tau}) \otimes \bar{\vec{n}} =
\alpha H \bar{\vec{n}} \otimes \bar{\vec{n}} \otimes \bar{\tau}$ on $\bar{I}$,
$(\bar\tau\cdot\nabla)\bar{\vec{n}}=-H\bar\tau$ on ${\bar{I}}$ by \eqref{eq:derivativeNormal},
as well as using \eqref{gradient_npp} and in the final step the choice \eqref{ODE_alpha},
the second derivative evaluated at the triple junction turns out to be
\begin{equation}\label{eq:xi_second_derivative_tj}
\begin{aligned}
		\nabla^2 \widetilde \xi(p)
		&~=\big((\bar\tau\cdot\nabla)\alpha - H^2\big)
		\bar\tau\otimes(\bar\tau\otimes\bar{\vec{n}}+\bar{\vec{n}}\otimes\bar\tau)
		\\&~~~~
		+\bar{\tau} \otimes \big(
		(- (\bar{\tau} \cdot \nabla ) H  -\alpha H) \, \bar{\tau} \otimes \bar{\tau}
		+ \beta \, \bar{\vec{n}} \otimes \bar{\vec{n}}
		\big) \\
		&~~~~ - \bar{\vec{n}}\otimes (H\bar\tau - \alpha\bar{\vec{n}}) \otimes (H\bar\tau - \alpha\bar{\vec{n}})
		\\&
		\stackrel{\eqref{ODE_alpha}}{=} \bar{\tau} \otimes \big(
		(- (\bar{\tau} \cdot \nabla ) H  -\alpha H) \, \bar{\tau} \otimes \bar{\tau}
		+ \beta \, \bar{\vec{n}} \otimes \bar{\vec{n}}
		\big) \\
		&~~~~ - \bar{\vec{n}}\otimes (H\bar\tau - \alpha\bar{\vec{n}}) \otimes (H\bar\tau - \alpha\bar{\vec{n}}).
\end{aligned}
\end{equation}

Now we are in a position to prove the two compatibility conditions
\eqref{compatibility_xi_first} and \eqref{compatibility_xi_second}.
We start with the former. By the identities \eqref{eq:xi_first_derivative_tj}
it is sufficient to check the two equalities
\begin{align}\label{eq:compatiblity_xi}
	- H_{1,2} \,  \bar{\tau}_{1,2}  + \alpha_{1,2} \,  \bar{\vec{n}}_{1,2} &  =
	- H_{2,3} \,  \bar{\tau}_{2,3}  + \alpha_{2,3} \,  \bar{\vec{n}}_{2,3}  =
	- H_{3,1} \,  \bar{\tau}_{3,1}  + \alpha_{3,1} \,  \bar{\vec{n}}_{3,1},
\end{align}
as \eqref{def_rotation} together with $J\bar\tau=\bar{\vec{n}}$ (cf.\ \eqref{eq:rotation})
imply $\bar\tau_{i,i+1}(p(t),t) = R_{(i,j)} \bar\tau_{j,j+1}(p(t),t)$ from which one can
then infer with \eqref{eq:compatiblity_xi} that
\begin{align*}
\nabla \big( R_{(i,j)} \widetilde \xi_{j,j+1} \big)
&= R_{(i,j)}\big(-H_{j,j+1}\bar{\tau}_{j,j+1}\otimes\bar\tau_{j,j+1}
+ \alpha_{j,j+1}\bar\tau_{j,j+1}\otimes\bar{\vec{n}}_{j,j+1}\big)
\\&
=\bar\tau_{i,i+1}\otimes\big(-H_{j,j+1}\bar\tau_{j,j+1}
+ \alpha_{j,j+1}\bar{\vec{n}}_{j,j+1}\big)
\\&
=\nabla \widetilde \xi_{i,i+1}
\end{align*}
at $(p(t),t)$ as asserted.

The identities \eqref{eq:compatiblity_xi} follow by using the evolution equation
$\frac{\mathrm{d}}{\mathrm{d}t} p(t) \cdot \bar{\vec{n}}_{i,i+1} = H_{i,i+1}$
at the triple junction to get
\begin{align}\label{eq:evolutionTripleJunction}
	\frac{\mathrm{d}}{\mathrm{d}t} p(t) = H_{i,i+1} \bar{\vec{n}}_{i,i+1}
	+ \Big( \bar{\tau}_{i,i+1} \cdot \frac{\mathrm{d}}{\mathrm{d}t} p(t) \Big) \bar{\tau}_{i,i+1}
\end{align}
for $i\in\{1,2,3\}$, identifying the term in the parenthesis as
$\alpha_{i,i+1}(p(t))$ by \eqref{ODE_alpha} and multiplying the
above equation with the rotation matrix $J$.

Turning to the second order compatibility condition, for $i\in\{1,2,3\}$
we use our precise choice \eqref{def_beta} for the coefficient functions at the triple junction
so that the second derivatives at the triple junction \eqref{eq:xi_second_derivative_tj} take the form
\begin{align*}
	\begin{split}
		\nabla^2 \widetilde \xi_{i,i+1}(p) & =\beta_{i,i+1}\, \bar{\tau}_{i,i+1} \otimes  \operatorname{Id} \\
		& \quad - \bar{\vec{n}}_{i,i+1} \otimes (H_{i,i+1}\bar{\tau}_{i,i+1}
		- \alpha_{i,i+1} \bar{\vec{n}}_{i,i+1}) \otimes (H_{i,i+1}\bar{\tau}_{i,i+1}
		- \alpha_{i,i+1} \bar{\vec{n}}_{i,i+1}).
	\end{split}
\end{align*}
Hence, because of \eqref{eq:compatiblity_xi}, \eqref{def_rotation} and
$\bar\tau_{i,i+1}(p(t),t) = R_{(i,j)} \bar\tau_{j,j+1}(p(t),t)$
it suffices to show that $\beta_{i,i+1}(p)$ is independent of $i$
in order to prove the second order compatibility condition \eqref{compatibility_xi_second}. However,
this is a consequence of differentiating in time the Herring angle condition \eqref{eq:auxHerring}.
Indeed, one may compute using \eqref{eq:derivativeNormal},
\eqref{ODE_alpha}, and \eqref{TransportNormalVectorTwoPhase}
\begin{equation}\label{eq:betaIndependent}
\begin{aligned}
0 &= \frac{\mathrm{d}}{\mathrm{d}t} \sum_{i=1}^3 \sigma_{i,i+1}\bar{\vec{n}}_{i,i+1}(p(t),t)
\\&
= \sum_{i=1}^3\sigma_{i,i+1}\nabla\bar{\vec{n}}_{i,i+1}(p(t),t)\frac{\mathrm{d}}{\mathrm{d}t} p(t)
+\sum_{i=1}^3\sigma_{i,i+1}\partial_t\bar{\vec{n}}_{i,i+1}(p(t),t)
\\&
=-\sum_{i=1}^3\sigma_{i,i+1}\bar{\tau}_{i,i+1}(p(t),t)
\alpha_{i,i+1}(p(t),t)H_{i,i+1}(p(t),t)
\\&~~~
-\sum_{i=1}^3\sigma_{i,i+1}\bar{\tau}_{i,i+1}(p(t),t)
\big(\bar\tau_{i,i+1}(p(t),t)\cdot\nabla\big)H_{i,i+1}(p(t),t)
\end{aligned}
\end{equation}
from which the claim is now immediate. This concludes the proof of Lemma~\ref{lemma:compatibility_xi}.
\end{proof}

Recall that apart from the family of vector fields $(\xi_{i,j})_{i\neq j}$, the notion of gradient flow calibrations
also requires a suitably defined velocity field $B$. For its construction in the vicinity
of a triple junction, we introduce in a first step certain auxiliary symmetric
velocity fields $\widetilde B_{(i,j)}=\widetilde B_{(j,i)}$.
To this end, we start for every $i\in\{1,2,3\}$ with an expansion ansatz of the form
\begin{equation}\label{AnsatzAuxiliaryVelocityHalfSpace}
\begin{aligned}
		\widetilde B_{(i,i+1)}(x,t)
		&:= H_{i,i+1}\big(P_{i,i+1}(x,t),t\big) \bar{\vec{n}}_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
		\\&~~~~
		+ \alpha_{i,i+1}\big(P_{i,i+1}(x,t),t\big) \bar{\tau}_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
		\\&~~~~
		+ \beta_{i,i+1}\big(P_{i,i+1}(x,t),t\big) s_{i,i+1}(x,t) \bar{\tau}_{i,i+1}\big(P_{i,i+1}(x,t),t\big)
\end{aligned}
\end{equation}
and $\widetilde B_{(i+1,i)} := \widetilde{B}_{(i,i+1)}$ for $t\in [0,T_{\mathrm{strong}})$ and $x \in \mathbb{H}_{i,i+1}(t)\cap B_r(p(t))$.
The definition of the coefficients $\alpha_{i,i+1}$ and $\beta_{i,i+1}$ is given in equations
\eqref{ODE_alpha} and \eqref{def_beta}.

\begin{lemma}
\label{lemma:regularity_tilde_B}
Let the assumptions and notation of Proposition~\ref{prop:xi_triple_junction} be in place.
For all phases $i\in\{1,2,3\}$, the auxiliary velocity field
$\widetilde B_{(i,i+1)}$ is of class $C^0_tC^2_x$ in the space-time domain
$\bigcup_{t\in [0,T_{\mathrm{strong}})}\big(\overline{\mathbb{H}}_{i,i+1}(t)\cap B_r(p(t))\big)\times\{t\}$
with bounded highest order derivatives. More precisely, we have the estimates
\begin{align}
\label{eq:boundsDerivativesTildeB}
|\nabla^k\widetilde B_{(i,i+1)}| \leq Cr_c^{-k-2},\quad k\in\{0,1,2\}.
\end{align}
Here, $r_c>0$ is the parameter from Definition~\ref{DefinitionSmoothlyEvolvingPartition}
of a smoothly evolving network of interfaces.
\end{lemma}

\begin{proof}
In the proof of Lemma~\ref{lemma:regularity_tilde_xi} we have seen
$s_{i,i+1} \in C^0_tC^4_x$ and $\bar{\vec{n}}_{i,i+1}, \bar{\tau}_{i,i+1} \in C^0_tC^3_x$,
as well as $H_{i,i+1} \in C^0_tC_x^3$, $\alpha_{i,i+1} \in C^0_tC^3_x$, and $\beta_{i,i+1} \in C^0_tC^2_x$.
The ansatz \eqref{AnsatzAuxiliaryVelocityHalfSpace} then implies the desired regularity
and the bound \eqref{eq:boundsDerivativesTildeB} follows from the estimates \eqref{BoundMeanCurvature} and
\eqref{BoundCoefficients}.
\end{proof}

We now have to make sure that our ansatz \eqref{AnsatzAuxiliaryVelocityHalfSpace}
for the auxiliary velocity fields satisfies first-order compatibility conditions
at the triple junction.

\begin{lemma}
\label{lemma:compatiblity_B}
Let the assumptions and notation of Proposition~\ref{prop:xi_triple_junction} be in place.
	For  every $i,j\in\{1,2,3\}$, the ansatz \eqref{AnsatzAuxiliaryVelocityHalfSpace} then satisfies
	\begin{align}
	\label{compatibility_B_zero}
	\widetilde B_{(i,i+1)}(p(t),t) &= \widetilde B_{(j,j+1)}(p(t),t)
	= \frac{\mathrm{d}}{\mathrm{d}t} p(t),
	\\
	\label{compatibility_B_first}
	\nabla\widetilde B_{(i,i+1)}(p(t),t) &= \nabla\widetilde B_{(j,j+1)}(p(t),t)
	\end{align}
	for all $t\in [0,T_{\mathrm{strong}})$.
\end{lemma}

\begin{proof}
  We fix again the index $i$ and omit all indices, superscripts,
	and arguments unless specifically required.	At the triple junction, we have
	\begin{align}
		\widetilde B(p(t),t) = \frac{\mathrm{d}}{\mathrm{d}t} p(t)
	\end{align}
	by the choice of the tangential coefficient $\alpha$, see \eqref{ODE_alpha},
	and the evolution equation $\frac{\mathrm{d}}{\mathrm{d}t} p(t) \cdot \bar{\vec{n}}_{i,i+1} = H_{i,i+1}$.
	This of course proves \eqref{compatibility_B_zero}.

	An explicit computation making use of the ansatz \eqref{AnsatzAuxiliaryVelocityHalfSpace},
	the identities \eqref{gradient_npp}, \eqref{eq:derivativeNormal} and \eqref{eq:derivativeTangent}
	as well as the choices of the coefficients \eqref{ODE_alpha} and \eqref{def_beta} moreover gives
	\begin{equation}\label{nabla_B_tilde}
	\begin{aligned}
		\nabla \widetilde B & = \left(- H^2
		+(\bar{\tau} \cdot \nabla  \alpha) \right)\bar{\tau}\otimes \bar{\tau}
		\\
		& \quad  +( (\bar{\tau}\cdot \nabla) H + \alpha H )
		\bar{\vec{n}}\otimes \bar{\tau}
		\\
		& \quad  + \beta \bar{\tau} \otimes \bar{\vec{n}} + O(|s|)
		\\
		& = \beta \left(\bar{\tau} \otimes \bar{\vec{n}}
		- \bar{\vec{n}} \otimes \bar{\tau} \right)+ O(|s|).
	\end{aligned}
	\end{equation}
	As we have $\left(\bar{\tau} \otimes \bar{\vec{n}} - \bar{\vec{n}} \otimes \bar{\tau}  \right) \bar{\vec{n}}
	= \bar{\tau} = J \bar{\vec{n}}$ and
	$\left(\bar{\tau} \otimes \bar{\vec{n}} - \bar{\vec{n}} \otimes \bar{\tau} \right) \bar{\tau}
	= - \bar{\vec{n}} = J \bar{\tau}$	it follows that	$\left(\bar{\tau} \otimes \bar{\vec{n}}
	- \bar{\vec{n}} \otimes \bar{\tau} \right) = J$, where $J$ was defined in \eqref{eq:rotation}. Therefore we get
	\begin{align}\label{short_nabla_B_tilde}
		\nabla \widetilde B & = \beta J	+ O(|s|).
	\end{align}
	Furthermore, by the computation in \eqref{eq:betaIndependent} and the definition \eqref{def_beta} we know
	that $\beta_{i,i+1}$ is independent of $i\in\{1,2,3\}$ at the triple junction.
	This entails the first-order compatibility condition \eqref{compatibility_B_first}.
\end{proof}

In a preparatory step towards the proof of \eqref{TransportEquationXiTriod} and \eqref{DissipTriod},
we now present the corresponding estimates for the (rotated) auxiliary vector fields $\widetilde \xi_{i,i+1}$
and the auxiliary velocity fields $\widetilde B_{(i,i+1)}$ on their respective domains of definition.

\begin{lemma}\label{EquationsTildeConstructions}
  Let the assumptions and notation of Proposition~\ref{prop:xi_triple_junction} be in place.
	Then there exists a constant $C>0$, depending only on the given regular triod
	$\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)$, such that for every $i,j\in\{1,2,3\}$,
	it holds throughout the space-time domain
	$\mathcal{U}_{j,j+1}:=\bigcup_{t\in[0,T_{\mathrm{strong}})}(\mathbb{H}_{j,j+1}(t)\cap B_r(p(t)))\times\{t\}$ that
	\begin{align}
		\label{TransportEquationXiTildeTriod}
		&\big|\partial_tR_{(i,j)}\widetilde\xi_{j,j+1}
		+ (\widetilde B_{(j,j+1)}\cdot\nabla)R_{(i,j)}\widetilde\xi_{j,j+1}
		+(\nabla\widetilde B_{(j,j+1)})^\mathsf{T}R_{(i,j)}\widetilde\xi_{j,j+1}\big|
		\\&\nonumber
		\leq C\dist(\cdot,{\bar{I}}_{j,j+1}),
		\\
		\label{DissipXiTildeTriod}
		&\big|\nabla\cdot R_{(i,j)}\widetilde\xi_{j,j+1}
		+\widetilde B_{(j,j+1)}\cdot R_{(i,j)}\widetilde\xi_{j,j+1}\big|
		\leq C\dist^2(\cdot,{\bar{I}}_{j,j+1}).
	\end{align}
	Moreover, we have for every $j\in\{1,2,3\}$
	\begin{align}
	  \label{LengthXiTildeTriod}
		\big|1-|\widetilde \xi_{j,j+1}|^2\big| &\leq  C\dist^3(\cdot,{\bar{I}}_{j,j+1}),
		\\
		\label{LengthConservationXiTildeTriod}
		\big|\partial_t|\widetilde\xi_{j,j+1}|^2
		+ (\widetilde B_{(j,j+1)}\cdot\nabla)|\widetilde\xi_{j,j+1}|^2 \big|
		&\leq  C\dist^2(\cdot,{\bar{I}}_{j,j+1})
	\end{align}
	throughout 	$\mathcal{U}_{j,j+1}$.
	We also have for all pairs $i,j\in\{1,2,3\}$ with $i\neq j$ that
	\begin{align}
	\label{boundCompatibilityXi}
	|R_{(i,j)}\widetilde\xi_{j,j+1}-R_{(i,j-1)}\widetilde\xi_{j-1,j}| &\leq
	C\dist^2(\cdot,p(t)),
	\\\label{boundCompatibilityGradientXi}
	|\nabla R_{(i,j)}\widetilde\xi_{j,j+1}-\nabla R_{(i,j-1)}\widetilde\xi_{j-1,j}| &\leq
	C\dist(\cdot,p(t))
	\end{align}
	in the intersection $\mathcal{U}_{i,i+1}\cap \mathcal{U}_{j,j+1}$
	Finally, for all $i,j\in\{1,2,3\}$ with $i\neq j$ it holds
	\begin{align}
	\label{boundCompatibilityVelocity}
	|\widetilde B_{(i,i+1)} - \widetilde B_{(j,j+1)}| &\leq C\dist^{2}(\cdot,p(t))
		\\\label{boundCompatibilityGradientVelocity}
	|\nabla \widetilde B_{(i,i+1)}-\nabla \widetilde B_{(j,j+1)}| &\leq
	C\dist(\cdot,p(t))
	\end{align}
	in the $\mathcal{U}_{i,i+1}\cap \mathcal{U}_{j,j+1}$.
\end{lemma}
\begin{proof}
  The coercivity estimates \eqref{LengthXiTildeTriod} and \eqref{LengthConservationXiTildeTriod}
	follow from a straightforward computation using the definition \eqref{AnsatzAuxiliaryXiHalfSpace}.
	Indeed, just note that because of \eqref{AnsatzAuxiliaryXiHalfSpace} it holds
	\begin{align}
	\notag|\widetilde\xi_{i,j}|^2 &= \Big(1-\frac{1}{2}\alpha_{i,j}^2s_{i,j}^2\Big)^2
	+ \Big(\alpha_{i,j}s_{i,j}+\frac{1}{2}\beta_{i,j}s_{i,j}^2\Big)^2
	\\&
	=1 + \alpha_{i,j}\beta_{i,j}s_{i,j}^3 + \frac{1}{4}(\alpha_{i,j}^4+\beta_{i,j}^2)s_{i,j}^4\label{eq:ComputeLengthOfXiTilde}
	\end{align}
	from which both \eqref{LengthXiTildeTriod} and \eqref{LengthConservationXiTildeTriod}
	follow immediately.

	To prove the estimate \eqref{TransportEquationXiTildeTriod}, let $i\in\{1,2,3\}$ be fixed.
	For what follows, let us slightly abuse notation by writing
	$\bar{\vec{n}}_{i,i+1}(x,t) = \bar{\vec{n}}_{i,i+1}(P_{i,i+1}(x,t),t),\,
	\alpha_{i,i+1}(x,t)=\alpha_{i,i+1}(P_{i,i+1}(x,t),t)$ and so on.
	Let us also omit all indices, superscripts and arguments unless specifically required.
	We first consider	the case $j=i$. Differentiating our ansatz \eqref{AnsatzAuxiliaryXiHalfSpace}
	for $\widetilde\xi$ in time using equation \eqref{TransportSignedDistanceTwoPhase}, i.e.,
 	$\partial_t s = - H$ on ${\bar{I}}$, and its extension \eqref{eq:alternativeReprScalarMeanCurvature}
	\begin{align*}
		\partial_t \widetilde \xi
		&=
		\nabla\partial_t s
		+ \alpha(\partial_t s)\bar\tau
		+ O(|s|)
		= - \nabla H - \alpha H\bar\tau + O(|s|).
	\end{align*}
	Moreover, we may compute by means of our ansatz
	\eqref{AnsatzAuxiliaryVelocityHalfSpace} for the auxiliary velocity field $\widetilde B$
	and the explicit computation \eqref{eq:xi_first_derivative} of $\nabla\widetilde\xi$ that
	\begin{align*}
		\big(\widetilde B \cdot\nabla\big) \widetilde \xi = O(|s|),
	\end{align*}
	and, resulting from our ansatz \eqref{AnsatzAuxiliaryXiHalfSpace} for $\widetilde\xi$ and
	the explicit computation \eqref{nabla_B_tilde} of $\nabla\widetilde B$
	that
	\begin{align*}
		&\big(\nabla\widetilde B\big)^\mathsf{T} \widetilde \xi
		=\big((\bar\tau\cdot\nabla)H + \alpha H\big)\bar\tau
		+O(|s|).
	\end{align*}
	As we have $\nabla H=\big((\bar\tau\cdot\nabla)H\big)\bar\tau$,
	the last three identities imply the desired bound \eqref{TransportEquationXiTildeTriod}
	on $\mathbb{H}_{i,i+1}\cap B_r(p)$.

The validity of \eqref{TransportEquationXiTildeTriod} for $j\neq i$ essentially
boils down to a commutator estimate. In the following, let us abbreviate
$\widetilde\xi=\widetilde\xi_{j,j+1}$,
$R=R_{(i,j)}$ as well as $\widetilde B=\widetilde B_{(j,j+1)}$. Then, by the definition
\eqref{AnsatzAuxiliaryXiHalfSpace} of $\widetilde\xi$ and exploiting the first case $j=i$ we deduce
\begin{align*}
&\partial_tR\widetilde\xi
+(\widetilde B\cdot\nabla)R\widetilde\xi
+(\nabla\widetilde B)^\mathsf{T}R\widetilde\xi
\\&
=R\big(\partial_t\widetilde\xi
+ (\widetilde B\cdot\nabla)\widetilde\xi
+ (\nabla\widetilde B)^\mathsf{T}\widetilde\xi\,\big)
+\big((\nabla\widetilde B)^\mathsf{T}R
- R(\nabla\widetilde B)^\mathsf{T}\big)\widetilde\xi
\\&
= O(\dist(\cdot,{\bar{I}}_{j,j+1})) + [(\nabla\widetilde B)^\mathsf{T},R]\widetilde\xi.
\end{align*}
By our computation \eqref{nabla_B_tilde} we have $\nabla\widetilde B = \beta J+ O(|s|)$.
Using the fact that $[J^\mathsf{T},R]=0$ on account of both matrices being rotations in the plane we get
\begin{align*}
[(\nabla\widetilde B)^\mathsf{T},R]
= \beta[J^\mathsf{T},R] + O(|s_{j,j+1}|)
= O(|s_{j,j+1}|).
\end{align*}
Therefore, we obtain \eqref{TransportEquationXiTildeTriod} also for all $j\neq i$
on $\mathbb{H}_{j,j+1}\cap B_r(p)$.

In the next step we check the validity of \eqref{DissipXiTildeTriod}. We again make use of the
abbreviations $\widetilde\xi=\widetilde\xi_{j,j+1}$, $R=R_{(i,j)}$ and $\widetilde B=\widetilde B_{(j,j+1)}$.
Due to the computation \eqref{eq:xi_first_derivative} of $\nabla \widetilde \xi$
we may compute on the one side
\begin{align*}
\nabla\cdot R\widetilde\xi
&= - H( R\bar\tau \cdot \bar\tau )
+(\alpha+\beta s)
(R\bar\tau \cdot  \vec{\bar{n}} )
+\alpha H s
(R\vec{\bar{n}} \cdot  \bar\tau )
-\alpha^2 s
(R\vec{\bar{n}} \cdot  \vec{\bar{n}} )
\\&~~~
+O(\dist^2(\cdot,{\bar{I}}_{j,j+1})).
\end{align*}
On the other side, making use of the
definitions \eqref{AnsatzAuxiliaryXiHalfSpace}
and \eqref{AnsatzAuxiliaryVelocityHalfSpace} of $\widetilde\xi$
and $\widetilde B$ we obtain
\begin{align*}
\widetilde B\cdot R\widetilde\xi
&= H\vec{\bar{n}}\cdot R\vec{\bar{n}}
+(\alpha + \beta s)
(\bar\tau\cdot R\vec{\bar{n}})
+\alpha H s
(\vec{\bar{n}}\cdot R\bar\tau)
+\alpha^2 s
(\bar\tau\cdot R\bar\tau)
\\&~~~
+O(\dist^2(\cdot,{\bar{I}}_{j,j+1})).
\end{align*}
Taking the last two identities together therefore yields the estimate \eqref{DissipXiTildeTriod}
by the equations (recall that $J\bar\tau=\bar{\vec{n}}$ and by the definition \eqref{eq:rotation} of $J$
that $J^T=J^{-1}=-J$)
\begin{align*}
	R\bar{\tau} \cdot \bar{\tau} & = R J^{-1} \bar{\vec{n}} \cdot \bar{\tau}
	= R \bar{\vec{n}} \cdot J\bar{\tau} = R \bar{\vec{n}} \cdot  \bar{\vec{n}},
	\\
	R\bar{\tau} \cdot \bar{\vec{n}}  & =  R J^{-1} \bar{\vec{n}} \cdot \bar{\vec{n}}
	= R \bar{\vec{n}} \cdot J\bar{\vec{n}} = - R \bar{\vec{n}} \cdot \bar{\tau}.
\end{align*}

We proceed with the verification of the bounds \eqref{boundCompatibilityXi} and \eqref{boundCompatibilityGradientXi}.
However, these are straightforward consequences of the compatibility conditions \eqref{compatibility_xi_zero},
\eqref{compatibility_xi_first} and \eqref{compatibility_xi_second} which allow by adding zero that
we can insert in $|R_{(i,j)}\widetilde\xi_{j,j+1}-R_{(i,j-1)}\widetilde\xi_{j-1,j}|$ the second-order
Taylor expansions based at $p(t)$ of both $R_{(i,j)}\widetilde\xi_{j,j+1}$ and $R_{(i,j-1)}\widetilde\xi_{j-1,j}$.
Together with \eqref{eq:boundsDerivativesTildeXi} the bound \eqref{boundCompatibilityXi} is now immediate.
One can argue similarly for the estimate \eqref{boundCompatibilityGradientXi}.

It remains to check \eqref{boundCompatibilityVelocity} and \eqref{boundCompatibilityGradientVelocity}. These bounds follow from the compatibility conditions
\eqref{compatibility_B_zero} and \eqref{compatibility_B_first}, adding and subtracting the second order Taylor
expansions based at the triple junction $p(t)$ of both $\widetilde B_{(i,i+1)}$ and $\widetilde B_{(j,j+1)}$ as well as
\eqref{eq:boundsDerivativesTildeB}. This concludes the proof of Lemma~\ref{EquationsTildeConstructions}.
\end{proof}

\subsection{Global construction by interpolation}\label{subsection:step_2}
Let $\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)$ be a regular triod
with triple junction $p(t)$ moving by mean curvature in the sense of Definition~\ref{def:triod}
on the space-time domain $\bigcup_{[0,T_{\mathrm{strong}})} B_{r}(p(t))\times\{t\}$.
As we discussed in the previous subsection, the auxiliary vector fields $\widetilde\xi_{i,i+1}$
and the auxiliary velocity field $\widetilde B_{(i,i+1)}$ serve as
the definition of the vector fields $\xi_{i,i+1}$ and the velocity field $B$ on
the wedge $W_{i,i+1}$, see Figure~\ref{fig:triod_wedges} for the partition
of the neighborhood of the triple junction.

The next step is to extend $\xi_{i,i+1}$ and $B$ to the entirety of
the space-time domain. As we want \eqref{eq:auxHerringAngleConditionExtensions}
to hold throughout the ball $B_r(p(t))$ we are essentially forced to set
$\xi_{i,i+1} = R_{(i,j)} \xi_{j,j+1}$ for all $i,j\in\{1,2,3\}$
wherever the latter is defined, and where $R_{(i,j)}$ is given in Lemma \ref{lemma:compatibility_xi}.
In order to resolve the problem of the domains of definitions, i.e., the half-spaces
$\mathbb{H}_i$ overlapping, we resort to an interpolation procedure on the
interpolation wedges $W_{i}$ (see again Figure~\ref{fig:triod_wedges}) both for $\xi_{i,i+1}$ and for $B$.
We similarly deal with the issue of combining the velocity fields $\widetilde B_{(i,i+1)}$ into a single field.
To this end, we first define suitable interpolation functions which move and rotate with the triod.

\begin{lemma}
\label{lemma:interpolation_functions}
  Let the assumptions and notation of Proposition~\ref{prop:xi_triple_junction} be in place.
	Recall in particular from Definition~\ref{def:triod} that we defined $\bar{\tau}_{i,j}(p(t),t) :=
	J^{-1}\bar{\vec{n}}_{i,j}(p(t),t)$ for $i,j \in \{1,2,3\}$ with $i\neq j$, where $J$ was defined
	in \eqref{eq:rotation},	and then have fixed by means of Lemma~\ref{lem:wedges} wedges
	$W_{i}(t)$ and $W_{i,i+1}(t)$ according to this choice of tangent vectors at the triple junction,
	see also Figure~\ref{fig:triod_wedges}.

	Then there exists a constant $C>0$, depending only on the given regular triod
	$\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)$, and interpolation functions
	\begin{align*}
		\lambda_i\colon \bigcup_{t\in [0,T_{\mathrm{strong}})}
		\big(B_{r}(p(t)) \cap \overline{W}_{i}(t)\big)\setminus\{p(t)\}\times\{t\} &\to [0,1]
	\end{align*}
	for every $i\in\{1,2,3\}$	which satisfy the following properties:
	\begin{itemize}[leftmargin=0.7cm]
		\item[i)] It holds that
							\begin{align}
							\label{LambdaLeftBoundary}
							\lambda_i(x,t) &= 0 \qquad\text{for}\quad
							x\in\big(\partial W_{i}(t)\cap\partial W_{i,i+1}(t)\big)\setminus\{p(t)\},
							\\
							\label{LambdaRightBoundary}
							\lambda_i(x,t) &= 1 \qquad\text{for}\quad
							x\in\big(\partial W_{i}(t)\cap\partial W_{i-1,i}(t)\big)\setminus\{p(t)\}.
							\end{align}
		\item[ii)] We have the estimates
			\begin{align}
				\label{boundslambda1}
				|\partial_t \lambda_i(x,t)| + |\nabla\lambda_i(x,t)| & \leq  C\frac{1}{|x-p(t)|},
				\\
				\label{boundslambda2}
				|\partial_t\nabla\lambda_i(x,t)| + |\nabla^2 \lambda_i (x,t)| &  \leq  C\frac{1}{|x-p(t)|^{2}}
			\end{align}
			for all $t\in [0,T_{\mathrm{strong}})$ and all $x \in \big(B_{r}(p(t)) \cap \overline{W}_{i}(t)\big) \setminus \{p(t)\}$.
			Furthermore, it holds
			\begin{align}
			  \label{first_derivative_lambda_vanishes_on_wedge}
				\nabla \lambda_i(x,t) & = 0,\quad \partial_t \lambda_i (x,t) = 0,
				\\
				\label{second_derivative_lambda_vanishes_on_wedge}
				\nabla^2 \lambda_i(x,t) & = 0, \quad \nabla \partial_t \lambda_i(x,t)  = 0

			\end{align}
			for all $t\in [0,T_{\mathrm{strong}})$ and all $x \in \big(B_r(p(t))\cap\partial W_i(t)\big) \setminus \{p(t)\}$.
		\item[iii)] We have a bound on the advective derivative
			\begin{align}\label{advectionlambda}
				\Big| \partial_t  \lambda_i(x,t) + \Big(\frac{\mathrm{d}}{\mathrm{d}t}
				p(t) \cdot \nabla\Big)\lambda_i(x,t) \Big| & \leq C
			\end{align}
			for all $t \in [0,T)$ and all $x \in  \big(B_{r}(p(t)) \cap \overline{W}_{i}(t)\big)\setminus\{p(t)\}$.
	\end{itemize}
\end{lemma}

\begin{proof}
Due to \eqref{def:interpolationWedge}, the interpolation wedge $W_i(t)$ is the interior of the conical hull
spanned by two unit vectors, say $v_i(t)$ and $w_i(t)$, whereas $W_{i,i+1}(t)$ is the interior
of the conical hull spanned by unit vectors $v_i(t)$ and $w_{i+1}(t)$ due to \eqref{def:wedgeInterface}.
In particular, we can represent
$\partial W_i(t) \cap \partial W_{i,i+1}(t) =  \{\gamma v_i(t)\colon \gamma\geq 0\}$
and $\partial W_i(t) \cap \partial W_{i-1,i}(t) = \{\gamma w_i(t)\colon \gamma\geq 0\}$.
As the vectors $v_i(t)$ and $w_i(t)$ can be expressed as a (fixed-in-time) linear combination of the unit-normals
$\bar{\vec{n}}_{i,j}(p(t),t)$ at the triple junction, we have the
bounds $\frac{\mathrm{d}}{\mathrm{d}t}v_i(t),\frac{\mathrm{d}}{\mathrm{d}t}w_i(t)\in C^0([0,T_{strong}))$.

Let $\theta_i\in (0,\frac{\pi}{2})$ denote the opening angle of the interpolation wedge $W_i$,
i.e., we have $\cos(\theta_i) = v_i(t)\cdot w_i(t)$. Note that
$\theta_i$ is time-independent and fully determined by the admissible matrix
of surface tensions $(\sigma_{i,j})_{i,j = 1,2,3}$ for the given regular
triod $\bar\chi=(\bar\chi_1,\bar\chi_2,\bar\chi_3)$.
Let $\widetilde\lambda\colon\Rd[]\to [0,1]$ be any smooth function such that
$\widetilde\lambda\equiv 0$ on $(-\infty,\frac{1}{3}]$ and
$\widetilde\lambda\equiv 1$ on $[\frac{2}{3},\infty)$.
We define
\begin{align*}
\lambda_i(x,t):=\widetilde\lambda\Bigg(\frac{1{-}v_i(t)\cdot\frac{x{-}p(t)}{|x{-}p(t)|}}{1{-}\cos\theta_i}\Bigg).
\end{align*}

Then the properties \eqref{LambdaLeftBoundary}--\eqref{second_derivative_lambda_vanishes_on_wedge}
are immediate consequences of the definitions
and the fact that $\frac{\mathrm{d}}{\mathrm{d}t}v_i(t),\frac{\mathrm{d}}{\mathrm{d}t}w_i(t)\in C^0([0,T_{strong}))$
as observed above. It remains to check the bound \eqref{advectionlambda} on the advective derivative.
To this end, we abbreviate $\lambda_i(x,t)=\bar\lambda_i\big(v_i(t)\cdot\frac{x{-}p(t)}{|x{-}p(t)|}\big)$
and simply compute
\begin{align*}
\partial_t\lambda_i(x,t)
&= -\bar\lambda'_i\frac{v_i(t)}{|x{-}p(t)|}\cdot\Big( \operatorname{Id} {-} \frac{x{-}p(t)}{|x{-}p(t)|}
\otimes \frac{x{-}p(t)}{|x{-}p(t)|}  \Big)
\frac{\mathrm{d}}{\mathrm{d}t}p(t)
+\bar\lambda'_i\frac{x{-}p(t)}{|x{-}p(t)|}\cdot\frac{\mathrm{d}}{\mathrm{d}t}v_i(t)
\\&
=-\Big(\frac{\mathrm{d}}{\mathrm{d}t}p(t) \cdot \nabla\Big)\lambda_i(x,t)
+\bar\lambda'_i\frac{x{-}p(t)}{|x{-}p(t)|}\cdot\frac{\mathrm{d}}{\mathrm{d}t}v_i(t)
\end{align*}
where $\bar\lambda'_i$ is evaluated at $v_i(t)\cdot\frac{x{-}p(t)}{|x{-}p(t)|}$.
From this, the last remaining claim \eqref{advectionlambda} immediately follows due to the bound
$\frac{\mathrm{d}}{\mathrm{d}t}v_i(t)\in L^\infty_t$.
\end{proof}

Equipped with these interpolating functions we are finally in the position
to prove the main result of this section.

\begin{proof}[Proof of Proposition~\ref{prop:xi_triple_junction}]
\textit{Step 1: Interpolation of the vector fields.}
We define (not yet normalized) extensions of the normal vector fields $\bar{\vec{n}}_{i,j}$
on $\bigcup_{t\in [0,T_{\mathrm{strong}})}B_r(p(t))\times\{t\}$ as follows:
\begin{align}
\label{def:xi_triple_junction}
  \bar\xi_{i,i+1}(x,t) :=
  	\begin{cases}
  		 R_{(i,j)} \widetilde \xi_{j,j+1}(x,t) & \text{ if } x \in W_{j,j+1}(t),\\
  		\!\begin{aligned}
	  		 & (1{-}\lambda_j(x,t)) R_{(i,j)} \widetilde \xi_{j,j+1}(x,t)\\
  			  &\quad +  \lambda_j(x,t)R_{(i,j-1)} \widetilde \xi_{j-1,j} (x,t)
  		 \end{aligned}
  		 & \text{ if } x \in \overline{W}_{j}(t),
  	\end{cases}
\end{align}
and $\bar\xi_{i+1,i} := -\bar\xi_{i,i+1}$ for $i\in\{1,2,3\}$.
The velocity field is given by
\begin{align}
\label{def:velocity_triple_junction}
	B(x,t) := \begin{cases}
		\widetilde B_{(j,j+1)}(x,t) & \text{ if } x \in W_{j,j+1}(t),\\
		\begin{aligned}
		& (1{-}\lambda_{j}(x,t))  \widetilde B_{(j,j+1)}(x,t) \\
		&\quad + \lambda_j(x,t) \widetilde B_{(j-1,j)}(x,t)
		\end{aligned}
		& \text{ if } x \in \overline{W}_{j}(t).
	\end{cases}
\end{align}
In the subsequent steps of the proof, we first establish all required properties
in terms of the vector fields $(\bar\xi_{i,j},B)$. Only in the penultimate step we
will choose the radius $\tilde r = \tilde r(\bar\chi) \leq r$ and define
unit-length vector fields $\xi_{i,j}$ by normalization of the vector fields
$\bar \xi_{i,j}$ defined in \eqref{def:xi_triple_junction} above. The last step
is then devoted to verify the required properties for the normalized vector
fields $\xi_{i,j}$.

\textit{Step 2: Regularity of $\bar\xi_{i,j}$ and $B$, the estimates \eqref{boundDerivativesXi}
and \eqref{boundDerivativeB}, and properties i)--iii).}
We first remark that the above definitions make sense due to the inclusions
$W_{i,i+1}(t) \subset \mathbb{H}_{i,i+1}(t)$ and $W_{i}(t) \subset \mathbb{H}_{i,i+1}(t)\cap \mathbb{H}_{i-1,i}(t)$
of Lemma \ref{lem:wedges}. Indeed, these inclusions are precisely what is needed so
that the building blocks $\widetilde\xi_{i,i+1}$ and $\widetilde B_{(i,i+1)}$ are only
evaluated on their domains of definition.

By the compatibility condition \eqref{compatibility_xi_zero} in Lemma~\ref{lemma:compatibility_xi}
for the auxiliary vector fields $\widetilde\xi_{j,j+1}$ at the triple junction,
as well as the conditions \eqref{LambdaLeftBoundary} and \eqref{LambdaRightBoundary}
from Lemma~\ref{lemma:interpolation_functions} for the interpolation functions,
the vector fields $\bar\xi_{i,j}$ are continuous. Similarly, their first and second derivatives are continuous across
the boundaries of the interpolation wedges
$\bigcup_{t\in [0,T_{\mathrm{strong}})}\big(\big(B_r(p(t))\cap \partial W_i(t)\big)\setminus\{p(t)\}\big)\times\{t\}$
by the properties \eqref{first_derivative_lambda_vanishes_on_wedge} and \eqref{second_derivative_lambda_vanishes_on_wedge}
of the interpolation functions.

Moreover, all spatial derivatives up to second order are uniformly bounded in $B_r(p(t))\setminus\{p(t)\}$
with the asserted estimate given by \eqref{boundDerivativesXi}. Indeed, in the wedges $W_{j,j+1}$ containing the interfaces
this follows from the estimates \eqref{eq:boundsDerivativesTildeXi}
and the definition \eqref{def:xi_triple_junction}. On the closure of the interpolation wedges $W_j$,
we first compute using the definition \eqref{def:xi_triple_junction}
\begin{align*}
\nabla\bar\xi_{i,i+1}
&= (1{-}\lambda_j) \nabla R_{(i,j)} \widetilde \xi_{j,j+1}
+ \lambda_j \nabla R_{(i,j-1)} \widetilde \xi_{j-1,j}
\\&
~~~ - (R_{(i,j)} \widetilde \xi_{j,j+1}{-}R_{(i,j-1)} \widetilde \xi_{j-1,j})\nabla\lambda_j,
\\
\nabla^2\bar\xi_{i,i+1} &= (1{-}\lambda_j) \nabla^2 R_{(i,j)} \widetilde \xi_{j,j+1}
+ \lambda_j \nabla^2 R_{(i,j-1)} \widetilde \xi_{j-1,j}
\\&~~~
- (\nabla R_{(i,j)} \widetilde \xi_{j,j+1}{-}\nabla R_{(i,j-1)} \widetilde \xi_{j-1,j})\nabla\lambda_j
\\&~~~
- (R_{(i,j)} \widetilde \xi_{j,j+1}{-}R_{(i,j-1)} \widetilde \xi_{j-1,j})\nabla^2\lambda_j.
\end{align*}
Now, the bound \eqref{boundDerivativesXi} follows from the controlled blow-up \eqref{boundslambda1} and \eqref{boundslambda2}
of the interpolation functions and the estimates \eqref{eq:boundsDerivativesTildeXi}, \eqref{boundCompatibilityXi}
as well as \eqref{boundCompatibilityGradientXi} for the auxiliary vector fields $\widetilde\xi_{j,j+1}$.
In total, this proves $\bar\xi_{i,j}\in L^\infty_tW^{2,\infty}_x$ together with \eqref{boundDerivativesXi}.
The other property $\bar\xi_{i,j}\in W^{1,\infty}_tL^\infty_x$ follows similarly
making use of Lemma~\ref{lemma:regularity_tilde_xi}, Lemma~\ref{lemma:compatibility_xi}
and Lemma~\ref{lemma:interpolation_functions} and the computation
\begin{align*}
\partial_t\bar\xi_{i,i+1} &= (1{-}\lambda_j) \partial_t R_{(i,j)} \widetilde \xi_{j,j+1}
+ \lambda_j \partial_t R_{(i,j-1)} \widetilde \xi_{j-1,j}
\\&~~~
- (R_{(i,j)} \widetilde \xi_{j,j+1}{-}R_{(i,j-1)} \widetilde \xi_{j-1,j})\partial_t\lambda_j
\quad\text{in } W_j.
\end{align*}

We proceed with the regularity of the velocity field $B$. First, by the compatibility condition
\eqref{compatibility_B_zero} in Lemma~\ref{lemma:compatiblity_B}
for the auxiliary velocity fields $\widetilde B_{(j,j+1)}$ at the triple junction,
as well as the conditions \eqref{LambdaLeftBoundary} and \eqref{LambdaRightBoundary}
from Lemma~\ref{lemma:interpolation_functions} for the interpolation functions,
the velocity field $B$ is continuous. The asserted bound \eqref{boundDerivativeB}
is a consequence of the definition \eqref{def:velocity_triple_junction},
the estimates \eqref{eq:boundsDerivativesTildeB} and \eqref{boundCompatibilityVelocity}
for the auxiliary velocity fields, the controlled blow-up \eqref{boundslambda1}
of the interpolation functions as well as the computation
\begin{align*}
\nabla B = (1{-}\lambda_{j})  \nabla\widetilde B_{(j,j+1)}
		+ \lambda_j \nabla\widetilde B_{(j-1,j)}
		+ (\widetilde B_{(j-1,j)}{-}\widetilde B_{(j,j+1)})\nabla\lambda_j
		\quad\text{in } W_j.
\end{align*}
This proves $B \in L^\infty_tW^{1,\infty}_x$ and the estimate \eqref{boundDerivativeB}.

For every $i\in\{1,2,3\}$, we obtain $\bar\xi_{i,i+1}(x,t) = \widetilde \xi_{i,i+1}(x,t) = \bar{\vec{n}}_{i,i+1}(x,t)$
for all $t \in [0,T_{\mathrm{strong}})$ and all $x \in {\bar{I}}_{i,i+1}(t)$ from the inclusion
${\bar{I}}_{i,i+1} \subset W_{i,i+1}(t) \cup \{ p(t)\}$ and the ansatz \eqref{AnsatzAuxiliaryXiHalfSpace},
taking care of property i); obviously except for the normalization condition away from the interfaces. The second property
$\bar\xi_{i,j} = - \bar\xi_{j,i}$ for $i,j \in \{1,2,3\}$ with $i \neq j$ holds by definition.
For every $j\in\{1,2,3\}$ we moreover have
\begin{align*}
	\sigma_{1,2} \bar\xi_{1,2} + \sigma_{2,3} \bar\xi_{2,3} + \sigma_{3,1}  \bar\xi_{3,1} \equiv
	\left( \sigma_{1,2} R_{(1,j)}  + \sigma_{2,3} R_{(2,j)}   + \sigma_{3,1} R_{(3,j)}
	\right) \widetilde \xi_{j,j+1} =0
\end{align*}
on $W_{j,j+1}(t)$ by the defining property \eqref{def_rotation} of the rotations $R_{(i,j)}$.
A similar argument ensures validity of \eqref{xi_compatibility} on the interpolation wedges $\overline{W}_{j}(t)$.

\textit{Step 3: Proof of the estimate \eqref{TransportEquationXiTriod} for $(\bar\xi_{i,j},B)$.}
By the skew-symmetry $\bar\xi_{i,j}=-\xi_{j,i}$, we only have to check the remaining properties for $j=i+1$.
Let $i\in\{1,2,3\}$ be fixed. First, let us remark that
the validity of \eqref{TransportEquationXiTriod} for the vector field $\bar\xi_{i,i+1}$
on the wedges $W_{j,j+1}$ follows from the estimate \eqref{TransportEquationXiTildeTriod}
and the definitions \eqref{def:xi_triple_junction} and \eqref{def:velocity_triple_junction}.
Hence, it remains to prove the bound \eqref{TransportEquationXiTriod} for the vector field $\bar\xi_{i,i+1}$
on each interpolation wedge $W_{j}$, $j\in\{1,2,3\}$.

To this end, let us fix $j\in\{1,2,3\}$ and recall that $|x-p(t)| \leq C \dist(\cdot,{\bar{I}}_{j,j+1})$ on $W_j$.
For the sake of readability, let us introduce the abbreviations, $\lambda=\lambda_j$, $R=R_{(i,j)}$, $R'=R_{(i,j-1)}$,
$\widetilde\xi=\widetilde\xi_{j,j+1}$, $\widetilde\xi'=\widetilde\xi_{j-1,j}$,
$\widetilde B=\widetilde B_{(j,j+1)}$ and $\widetilde B'=\widetilde B_{(j-1,j)}$.
Using the product rule and the definition \eqref{def:xi_triple_junction} of $\bar\xi_{i,i+1}$ on the interpolation wedge $W_j$,
	\begin{align*}
		\left(\partial_t+ (B\cdot \nabla) + (\nabla B)^\mathsf{T} \right)\bar \xi_{i,i+1}
		=
		& (1-\lambda) \left( \partial_t + (B\cdot \nabla) + (\nabla B)^\mathsf{T}  \right) R \tilde \xi \\
		&+ \lambda  \left( \partial_t + (B\cdot \nabla) + (\nabla B)^\mathsf{T}  \right) R'\tilde \xi'
		\\& +\left(\partial_t \lambda + (B\cdot \nabla) \lambda \right)(R'\widetilde \xi' - R \widetilde \xi).
	\end{align*}
	Observe that by the bounds \eqref{boundslambda1} on $\lambda$ and the second-order compatibility \eqref{boundCompatibilityXi}, the last right-hand side term is of the desired order $O(|x-p(t)|)$.
	We want to manipulate the first two right-hand side terms to make the advection-equations \eqref{TransportEquationXiTildeTriod} appear. To this end, we write $B=\widetilde B + \lambda (\widetilde B' - \widetilde B)$ and obtain
	\begin{align*}
		 \left( \partial_t + (B\cdot \nabla) + (\nabla B)^\mathsf{T}  \right) R \tilde \xi
		 =
		 &  \left( \partial_t + (\widetilde B\cdot \nabla) + (\nabla \widetilde B)^\mathsf{T}  \right) R \tilde \xi
		 \\&- \left(\lambda (\widetilde B' - \widetilde B) \cdot \nabla\right) R\widetilde \xi
		 - \left((\widetilde B' - \widetilde B)\cdot R\widetilde \xi \right) \nabla \lambda
		 \\&- \lambda \left(\nabla (\widetilde B' - \widetilde B)\right)^\mathsf{T} R\widetilde \xi
	\end{align*}
	Using the compatibility conditions \eqref{boundCompatibilityXi}--\eqref{boundCompatibilityGradientVelocity}  alongside with the uniform bounds \eqref{eq:boundsDerivativesTildeXi}, \eqref{eq:boundsDerivativesTildeB}, and \eqref{boundslambda1} shows that the two three right-hand side terms are negligible.
	Indeed, for example \eqref{boundCompatibilityVelocity} and \eqref{eq:boundsDerivativesTildeXi} imply $|(\lambda (\widetilde B' - \widetilde B) \cdot \nabla) R\widetilde \xi| \leq |\widetilde B' -\widetilde B| |\nabla \widetilde \xi| \leq C |x-p(t)|^2$. Along the same lines, one shows that the other two terms are $O(|x-p(t)|)$.

	Arguing similarly via $B= \widetilde B' + (1-\lambda) (\widetilde B-\widetilde B')$ for the differential operator acting on $R'\widetilde \xi'$, we obtain
	\begin{align*}
		\left(\partial_t+ (B\cdot \nabla) + (\nabla B)^\mathsf{T} \right)\bar \xi_{i,i+1}
	=
	 &(1-\lambda) \left( \partial_t + (\widetilde B\cdot \nabla) + (\nabla \widetilde B)^\mathsf{T}  \right) R \tilde \xi \\
	&+ \lambda  \left( \partial_t + (\widetilde B'\cdot \nabla)
	+ (\nabla \widetilde B')^\mathsf{T}  \right) R'\tilde \xi' +O(|x-p(t)|)
	\end{align*}
	throughout $W_j$. By \eqref{TransportEquationXiTildeTriod}, the two transport equations on the right-hand side are satisfied to first order, which proves \eqref{TransportEquationXiTriod}.

\textit{Step 4: Proof of the estimate \eqref{DissipTriod} for  $(\bar\xi_{i,j},B)$.} Let $i\in\{1,2,3\}$ be fixed.
Note that because of \eqref{def:xi_triple_junction} as well as \eqref{DissipXiTildeTriod},
it only remains to prove \eqref{DissipTriod} for the vector
field $\bar\xi_{i,i+1}$ in the interpolation wedges $W_{j}$, $j\in\{1,2,3\}$.
We again fix $j\in\{1,2,3\}$ and abbreviate for the sake of readability
$\lambda=\lambda_j$, $R=R_{(i,j)}$, $R'=R_{(i,j-1)}$,
$\widetilde\xi=\widetilde\xi_{j,j+1}$, $\widetilde\xi'=\widetilde\xi_{j-1,j}$,
$\widetilde B=\widetilde B_{(j,j+1)}$ as well as $\widetilde B'=\widetilde B_{(j-1,j)}$.

Now, we proceed similarly as in the proof of \eqref{TransportEquationXiTriod}.
More precisely, making use of the definition \eqref{def:xi_triple_junction} we get
\begin{align*}
\nabla\cdot\bar\xi_{i,i+1} &=
\big((R'\widetilde\xi'{-}R\widetilde\xi)\cdot\nabla\big)\lambda
+(1{-}\lambda)\nabla\cdot R\widetilde\xi+\lambda\nabla\cdot R'\widetilde\xi'.
\end{align*}
By the controlled blow-up \eqref{boundslambda1} of the interpolation functions,
the estimate \eqref{boundCompatibilityXi} and the approximate mean curvature flow equation \eqref{DissipXiTildeTriod} it then follows
\begin{align*}
\nabla\cdot\bar\xi_{i,i+1} &=-(1-\lambda)\widetilde B\cdot R\widetilde\xi
-\lambda \widetilde B'\cdot R'\widetilde\xi'
+O(|x-p(t)|).
\end{align*}
Finally, the estimates \eqref{boundCompatibilityGradientXi} and \eqref{boundCompatibilityVelocity} in conjunction with definitions
\eqref{def:xi_triple_junction} and \eqref{def:velocity_triple_junction} imply the desired bound
\begin{align}
\label{eq:dissipTriodNoNormalization}
\nabla\cdot\bar\xi_{i,i+1} &= -B\cdot\bar\xi_{i,i+1} + O(|x-p(t)|).
\end{align}

\textit{Step 5: Proof of the estimate}
\begin{align}
\label{LengthXiTriod}
\big|1-|\bar\xi_{i,j}|^2\big| &\leq C\dist^2(x,{I}_{i,j}(t)) \quad
\text{ for all } (x,t) \in \bigcup_{[0,T_{\mathrm{strong}})} B_{r}(p(t))\times\{t\}.
\end{align}

Let $i\in\{1,2,3\}$ be fixed. The validity of \eqref{LengthXiTriod}
for the vector field $\bar\xi_{i,i+1}$ in the wedge $W_{i,i+1}$ follows from
\eqref{LengthXiTildeTriod}. Since on the wedges $W_{j,j+1},\,j\neq i,$
the vector field $\bar\xi_{i,i+1}$ is obtained from the vector field $\bar\xi_{j,j+1}$ by a mere rotation,
see \eqref{def:xi_triple_junction}, we also immediately obtain \eqref{LengthXiTriod}
for all $x\in W_{j,j+1}(t),\,j\neq i$.
Finally, note that on the interpolation wedges, since each of the two building blocks $\widetilde\xi_{j,j+1}$ and $\widetilde \xi_{j-1,j}$ satisfy \eqref{LengthXiTriod}, also their convex combination $\bar\xi_{i,i+1}$ does.

\textit{Step 6: Choice of $\tilde r = \tilde r(\bar\chi) \leq r$ and definition
of normalized vector fields $\xi_{i,j}$.}
By the definition \eqref{def:xi_triple_junction} of the vector fields $\bar\xi_{i,j}$ we have
$|\bar\xi_{i,j}(\cdot,t)|=1$ on the union of interfaces
${\bar{I}}(t)$ for all $t\in [0,T_{\mathrm{strong}})$. Due to their continuity and the
regularity assumptions on the moving triod in Definition~\ref{def:triod}, we may
choose a radius $\tilde r = \tilde r(\bar\chi) \leq r$ such that
$B_{\tilde r}(p(t)) \subset \{|\bar\xi_{i,j}(\cdot,t)|>\frac{1}{2}\}$ for all $t\in [0,T_{\mathrm{strong}})$
and all $i,j\in\{1,2,3\}$ with $i\neq j$. We then define
\begin{align}
\label{def:normalizedXi}
\xi_{i,j}(x,t) := \frac{\bar\xi_{i,j}(x,t)}{|\bar\xi_{i,j}(x,t)|} \quad
\text{for all } (x,t)\in\bigcup_{t\in[0,T_{\mathrm{strong}})}B_{\tilde r}(p(t)){\times}\{t\}
\end{align}
all $i,j\in\{1,2,3\}$ with $i\neq j$.
It remains to verify the asserted properties in terms of the vector fields $(\xi_{i,j},B)$
on the restricted space-time domain with radius $\tilde r$.

\textit{Step 7: Conclusion.}
Since $\xi_{i,j}(x,t)=\bar\xi_{i,j}(x,t)$ for all $t\in [0,T_{\mathrm{strong}})$
and all $x\in {\bar{I}}_{i,j}(t)$, property \textit{i)} is an immediate
consequence of definition \eqref{def:normalizedXi}. Obviously, the
skew-symmetry relation in property \textit{ii)} carries over from
$\bar\xi_{i,j}$ to $\xi_{i,j}$. Validity of the Herring angle condition
\eqref{xi_compatibility} in terms of the vector fields $\xi_{i,j}$
also follows immediately from their definition \eqref{def:normalizedXi}
and the fact that the vector fields $\bar\xi_{i,j}$ already
satisfy \eqref{xi_compatibility}. Indeed, just recall that the vector fields
$\bar\xi_{1,2},\,\bar\xi_{2,3}$ resp.\ $\bar\xi_{3,1}$ can be obtained from
each of the other ones by a rotation. The required regularity estimate
\eqref{boundDerivativesXi} is satisfied by the choice of the radius $\tilde r$
in the previous step, the definition \eqref{def:normalizedXi} of $\xi_{i,j}$
and the fact that the vector fields $\bar\xi_{i,j}$ are already subject to such an estimate
(as established in the second step of this proof). It therefore
remains to argue that the estimates \eqref{TransportEquationXiTriod}
and \eqref{DissipTriod} hold true.

Using the product rule, we may compute on $B_{\tilde r}(p(t))$ by means of the choice of $\tilde r$ in the previous step
\begin{align*}
&\left( \partial_t + (B\cdot \nabla) +(\nabla B)^\mathsf{T}\right) \frac{\bar\xi_{i,j}}{|\bar\xi_{i,j}|}\\
&~~=\frac1{|\bar\xi_{i,j}|} \left( \partial_t + (B\cdot \nabla) +(\nabla B)^\mathsf{T}\right)\bar\xi_{i,j}
-\frac1{2|\bar \xi_{i,j}|^3} \bar\xi_{i,j} \left( \partial_t + (B\cdot \nabla) \right) |\bar \xi_{i,j}|^2
\end{align*}
By Step 3, the first right-hand side term is of the order $O(|x-p(t)|)$. The same rate also holds for the second right-hand side term simply by multiplication with $\bar \xi_{i,j}$. This proves the estimate \eqref{TransportEquationXiTriod}.
We now turn to the proof of \eqref{DissipTriod}.
Here, we compute on $B_{\tilde r}(p(t))$ by means of the choice of $\tilde r$ in the
previous step
\begin{align*}
\nabla\cdot\frac{\bar\xi_{i,j}}{|\bar\xi_{i,j}|}
&=\frac{\nabla\cdot\bar\xi_{i,j}}{|\bar\xi_{i,j}|}
-\frac{(\bar\xi_{i,j}\cdot\nabla)|\bar\xi_{i,j}|^2}{2|\bar\xi_{i,j}|^3}.
\end{align*}
It is immediate from the identity \eqref{eq:ComputeLengthOfXiTilde} and the definition \eqref{def:xi_triple_junction} to estimate the second term. Using the approximate mean curvature flow equation \eqref{eq:dissipTriodNoNormalization} for the first term then yields
\begin{align*}
	\nabla\cdot\frac{\bar\xi_{i,j}}{|\bar\xi_{i,j}|}
	= -B\cdot\frac{\bar\xi_{i,j}}{|\bar\xi_{i,j}|}
	+ O\big(\dist(x,{\bar{I}}(t)) \big),
\end{align*}
which be definition \eqref{def:normalizedXi} of $\xi_{i,j}$ is nothing else than \eqref{DissipTriod}.
This concludes the proof of Proposition~\ref{prop:xi_triple_junction}.
\end{proof}

Finally, we provide the elementary-geometric proof for the existence of wedges
with the desired properties.

\begin{proof}[Proof of Lemma \ref{lem:wedges}]
	By rotation symmetry, we may without loss of generality assume for example $\tau_{1,2} =e_1$.
	Then the other two vectors $\tau_{2,3}$ and $\tau_{3,1}$ are determined up to a permutation by the requirements
	\begin{align}\label{wedges_balance_of_forces}
		\sigma_{1,2}\tau_{1,2} + \sigma_{2,3} \tau_{2,3} + \sigma_{3,1} \tau_{3,1} & = 0,
		\\\nonumber
		|\tau_{2,3}|^2 = |\tau_{3,1}|^2 & =1.
	\end{align}
	Therefore it is sufficient to prove the statement for a single choice of unit vectors
	$\tau_{1,2}$, $\tau_{2,3}$ and $\tau_{3,1}$. We may furthermore choose $p=0$.

	\begin{figure}
	  \begin{tikzpicture}[scale=3]
	  	\tikzmath{\l = .6;\L=.9;};
	  	\begin{scope}
	  		 \clip (0,0) circle [radius = .8];
	  		 \begin{scope}
	 			\clip (0,0) -- (40:1.5) --(90:1.5) -- (0,0);
	 			\foreach \i in {0,...,15}
				{
					\draw[color=blue] (-.8,\i*.07) -- (.8,\i*.07);
					\draw[color=red, opacity=.6] (-.8,\i*.07+.035 ) -- (.8,\i*.07 +.035);
				};
	 		\end{scope}

	 		\begin{scope}
	 			\clip (0,0) -- (156:1.5) --(184:1.5) -- (0,0);
	 			\foreach \i in {0,...,15}
				{
					\draw[opacity=.4] ($(120:1.5) + (210:\i*.07)$) -- ($(120:-1.5) + (210:\i*.07)$);
					\draw[color=red, opacity=.6] ($(120:1.5) + (210:\i*.07+.035)$) -- ($(120:-1.5) + (210:\i*.07+ .035)$);
				};
	 		\end{scope}

	 		\begin{scope}
	 			\clip (0,0) -- (270:1.5) --(300:1.5) -- (0,0);
	 			\foreach \i in {0,...,15}
				{
					\draw[opacity=.3] ($(240:1.5) + (330:\i*.07)$) -- ($(240:-1.5) + (330:\i*.07)$);
					\draw[color=blue] ($(240:1.5) + (330:\i*.07+.035)$) -- ($(240:-1.5) + (330:\i*.07+ .035)$);
				};
	 		\end{scope}

	  	\draw[-Latex,color=blue] (0,0) -- (0:\l);
	  	\node at ($(0:\l+.1)$) {$\tau_{3,1}$};
	  	\draw[-Latex,color=red,opacity=.6] (0,0) -- (130:\l);
	  	\node at ($(130:\l+.05)$) {$\tau_{1,2}$};
	  	\draw[-Latex,opacity=.4] (0,0) -- (210:\l);
	  	\node at ($(210:\l+.1)$) {$\tau_{2,3}$};

		\draw[dashed,color=red,opacity=.6] ($(40:-\L)$)--(40:0);

		\draw[dashed,opacity=.4] (120:0)--(120:\L);

		\draw[color=red,opacity=.6,thick] (0,0) -- (40:\L);
		\draw[color=blue,thick] (0,0) -- (90:\L);

		\draw[thick] (0,0) -- (156:1);
		\draw[thick] (0,0) -- (184:1);

		\draw[opacity=.4,thick] (0,0) -- (300:\L);
		\draw[color=blue,thick] (0,0) -- (270:\L);

		\end{scope}

		\node at  ($(285:\L)$)  {$W_{3}$};
		\node at ($(65:\L)$) {$W_{1}$};
		\node at ($(170:\L)$) {$W_{2}$};

		\node at (105:.9) {$W_{1,2}$};
		\node at (235:.9) {$W_{2,3}$};
		\node at (350:.9) {$W_{3,1}$};
	  \end{tikzpicture}
	  \caption{If the angle between two tangent vectors is less than $90^\circ$, we trisect it to obtain the desired interpolation wedge, see for example $W_2$. Otherwise, we take the corresponding intersection of the half-spaces, as is done for $W_1$ and $W_3$. The wedges $W_{1,2}$, $W_{2,3}$ and $W_{3,1}$ lie inbetween. \label{fig:wedges_proof}}
	\end{figure}

	Using the balance of forces condition \eqref{wedges_balance_of_forces} we see
	that there exist $\theta_i\in (0,\pi)$ such that $\cos(\theta_i) = \tau_{i,i+1} \cdot \tau_{i-1,i}$ for $i=1,2,3$.
	If $\theta_i > \frac{\pi}{2}$ we may simply define $v_i, w_i \in \mathbb{S}^1$ such that
	$W_i := \{\gamma_1 v_i + \gamma_2 w_i: \gamma_1,\gamma_2 \in (0,\infty) \} = \mathbb{H}_{i,i+1}\cap \mathbb{H}_{i-1,i}$.
	Otherwise, we choose them such that $W_i := \{\gamma_1 v_i + \gamma_2 w_i: \gamma_1,\gamma_2 \in (0,\infty) \}$
	is the middle third of the wedge	$\{\gamma_1 \tau_{i,i+1} + \gamma_2 \tau_{i-1,i}: \gamma_1,\gamma_2 \in (0,\infty)\}$.
	The desired properties then easily follow.
\end{proof}

\section{Gradient flow calibrations for a regular network}
\label{sec:networkConstruction}
The aim of this section is to prove Proposition~\ref{PropositionExistenceCalibration}: Given a strong solution to multiphase mean curvature flow (in the sense of an evolving network of smooth curves meeting at triple junctions), we construct a gradient flow calibration by gluing the local constructions from the previous two chapters.

In order to define these vector fields, we will distinguish
between the two distinct topological features being present in the network of interfaces
of a strong solution, namely triple junctions and smooth two-phase interfaces.
The global definitions will be obtained by gluing together suitable local definitions
$\xi^k_{i,j}$ and $B^k$ of these vector fields for each topological feature $\mathcal{T}_k$ by means
of a suitably defined partition of unity. The definition of such a partition of unity
and its localization properties are the content of Section~\ref{SectionPartitionOfUnity}.

We then proceed with the global definitions of the vector fields $\xi_{i,j}$
and $B$ in Section~\ref{SectionGlobalDefinitions}. The main building blocks are
given by the already mentioned local vector fields $\xi^k_{i,j}$ and $B^k$.
The construction of these vector fields together with a derivation of their main properties
was carried out in Section~\ref{SectionLocalConstructionsTwoPhase} for the model problem
of a smoothly evolving two-phase interface moving by mean curvature; or Section~\ref{SectionLocalConstructionsTriod} for
the model problem of a smoothly evolving triod moving by mean curvature. The main focus of
Section~\ref{SectionGlobalDefinitions} then consists of the question of the proper definition
of the vector fields $\xi_{i,j}^k$ in the case where at least either phase $i$ or $j$ are not locally
present at the selected topological feature $\mathcal{T}_k$.

Being equipped with the global definitions of the vector fields $\xi_{i,j}$ and $B$,
we proceed with the discussion of the compatibility between the local constructions
for the different topological features, see Section~\ref{SectionCompatibilityBoundsGluing}.
We use these bounds in Section~\ref{SectionBoundsTimeEvolutionGlobalConstruction}
to obtain the desired bounds on the time evolution of $\xi_{i,j}$ resp.\ $|\xi_{i,j}|^2$
as well as the validity of $\nabla\cdot\xi_{i,j}=- B\cdot\xi_{i,j}$ up to an error
being controlled by our relative entropy functional.

\subsection{Localization of topological features}\label{SectionPartitionOfUnity}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution} on some time interval $[0,T_{\mathrm{strong}})$.
In particular, the family $\bar\chi(t)$ is a smoothly evolving regular partition
and the family $({\bar{I}}_{i,j}(t))_{i\neq j}$ is a smoothly evolving regular network of interfaces
in the sense of Definition~\ref{DefinitionSmoothlyEvolvingPartition}.

We decompose the network of interfaces of the strong solution according to its topological features,
i.e., into smooth two-phase interfaces on the one hand and triple junctions on the other hand.
Suppose that the strong solution has $K$ of such topological features $\mathcal{T}_k$, $1\leq k\leq K$.
We then split $\{1,\ldots,K\}=:\mathcal{K}^{\mathrm{2ph}}\cup\mathcal{K}^{\mathrm{3j}}$ with the convention that $\mathcal{K}^{\mathrm{3j}}=\{1,\ldots,|\mathcal{K}^{\mathrm{3j}}|\}$,
and where $\mathcal{K}^{\mathrm{3j}}$ has the interpretation of enumerating the triple junctions $p_1,\ldots,p_{|\mathcal{K}^{\mathrm{3j}}|}$
present in the strong solution whereas $\mathcal{K}^{\mathrm{2ph}}$ enumerates the connected components (in space-time) of the
smooth two-phase interfaces ${\bar{I}}_{i,j}$. We next define $\mathcal{T}_k:=\{p_{k}\}$,
if $k\in\mathcal{K}^{\mathrm{3j}}$, as well as $\mathcal{T}_k\subset {\bar{I}}_{i,j}$ for the corresponding
space-time connected component $k\in\mathcal{K}^{\mathrm{2ph}}$ of a two-phase interface ${\bar{I}}_{i,j}$.
We say that the $i$-th phase of the strong solution is
\emph{present at the topological feature $\mathcal{T}_{k}$}
if $\partial\{\bar\chi_{i}=1\}\cap\mathcal{T}_{k}\neq\emptyset$.
Otherwise, we say that the phase is \emph{absent at $\mathcal{T}_k$}.

We now introduce a partition of unity $(\eta,\eta_1,\ldots,\eta_K)$, where each $\eta_k$
localizes in a neighborhood of the corresponding topological feature $\mathcal{T}_k$, as follows:

\begin{lemma}\label{LemmaPartitionOfUnity}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution},
whose network of interfaces decomposes into $K$ topological features $\mathcal{T}_k$.
Let $r_c$ be the associated constant from the definition of a smoothly evolving network
of interfaces, see Definition~\ref{DefinitionSmoothlyEvolvingPartition}.
Then, for every $k\in\{1,\ldots,K\}$ there exists a function
\begin{align*}
\eta_k \in L^\infty([0,T_{\mathrm{strong}});W^{2,\infty}(\Rd[2];[0,1])) \cap
W^{1,\infty}([0,T_{\mathrm{strong}});W^{1,\infty}(\Rd[2];[0,1]))
\end{align*}
such that the family $(\eta_1,\ldots,\eta_K)$ is a partition of unity in
the following sense:
\begin{itemize}[leftmargin=0.7cm]
\item[i)] Ket $\eta:=1-\sum_{k=1}^K\eta_k$. Then $\eta(x,t)\in [0,1]$
					for all $(x,t)\in\Rd[2]\times [0,T_{\mathrm{strong}})$. Moreover, on the evolving network of interfaces
					$\bigcup_{t\in [0,T_{\mathrm{strong}})}\bigcup_{i\neq j}{\bar{I}}_{i,j}(t)\times\{t\}$
					it already holds $\sum_{k=1}^K\eta_k\equiv 1$.
\end{itemize}
Furthermore, there exists a constant $0<\widetilde r_c\leq \frac{r_c}{2}$, such that
the family $(\eta_1,\ldots,\eta_K)$ is subject to the following localization properties:
\begin{itemize}[leftmargin=0.7cm]
\item[ii)] For all $k\in\mathcal{K}^{\mathrm{2ph}}$ we have
					 \begin{align}\label{LocalizationTwoPhase}
					 \supp\eta_k(\cdot,t)\subset\mathcal{T}_k(t)+B_{\widetilde r_c}(0).
					 \end{align}
					 Moreover, $\mathcal{T}_k$ is a regular interface evolving
					 by mean curvature in the space-time domain
					 $\bigcup_{t\in[0,T_{\mathrm{strong}})}\big(\mathcal{T}_k(t){+}B_{2\widetilde r_c}(0)\big)\times\{t\}$
					 in the precise sense of Definition~\ref{DefinitionStrongSolutionTwoPhase}.
\item[iii)] For all $k\in\mathcal{K}^{\mathrm{3j}}$ it holds
						\begin{align}\label{LocalizationTripleJunction}
						\supp\eta_k(\cdot,t)\subset B_{\widetilde r_c}(\mathcal{T}_k(t)).
						\end{align}
						Let $l_1,l_2,l_3\in\mathcal{K}^{\mathrm{2ph}}$ be the interfaces present at the triple junction $\mathcal{T}_k$.
						The three interfaces $(\mathcal{T}_{l_1},\mathcal{T}_{l_2},\mathcal{T}_{l_3})$ form
						a regular triod with triple junction $\mathcal{T}_{k}$
						evolving by mean curvature in the space-time domain
						$\bigcup_{t\in [0,T_{\mathrm{strong}})}B_{2\widetilde r_c}(\mathcal{T}_k(t))\times\{t\}$
						in the precise sense of Definition~\ref{def:triod} (with $r=2\widetilde r_c$).
\item[iv)] Let $k,k'\in\mathcal{K}^{\mathrm{3j}}$ be two distinct triple junctions. Then
				   \begin{align}\label{LocalizationTwoTripleJunctions}
					 \supp\eta_k(\cdot,t)\cap\supp\eta_{k'}(\cdot,t)\subset
					 B_{2\widetilde r_c}(\mathcal{T}_k(t))\cap B_{2\widetilde r_c}(\mathcal{T}_{k'}(t))
					 =\emptyset.
					 \end{align}
\item[v)] Let $k\in\mathcal{K}^{\mathrm{3j}}$ be a triple junction and let $l\in\mathcal{K}^{\mathrm{2ph}}$ be
					a two-phase interface. Then $\supp\eta_k\cap\supp\eta_l\neq\emptyset$ if and only if the interface
					$\mathcal{T}_{l}$ has an endpoint at the triple junction $\mathcal{T}_{k}$.
					In this case, it holds (with the notation of Definition~\ref{def:triod} in place)
					assuming that $\mathcal{T}_l\subset {\bar{I}}_{i,j}$
					\begin{align}\label{LocalizationTripleJunctionTwoPhase}
					\supp\eta_k(\cdot,t)\cap\supp\eta_l(\cdot,t)
					\subset B_{\widetilde r_c}(\mathcal{T}_k(t))\cap(W_{i,j}(t)\cup W_{i}(t)\cup W_{j}(t)).
					\end{align}
\item[vi)] Let $l,m\in\mathcal{K}^{\mathrm{2ph}}$ be two distinct two-phase interfaces. Then it holds
					 $\supp\eta_l\cap\supp\eta_m\neq\emptyset$ if and only if both interfaces
					 have an endpoint at the same triple junction $\mathcal{T}_{k},\,k\in\mathcal{K}^{\mathrm{3j}}$.
					 In this case, it holds (with the notation of Definition~\ref{def:triod} in place)
					 \begin{align}\label{LocalizationTwoInterfaces}
					 \supp\eta_l(\cdot,t)\cap\supp\eta_m(\cdot,t)\subset \supp\eta_k(\cdot,t)\cap W_{i}(t),
					 \end{align}
					 where we assume that $\mathcal{T}_{l}\subset {\bar{I}}_{i,j}$ and $\mathcal{T}_{m}\subset {\bar{I}}_{i,p}$.
\end{itemize}
Finally, the following coercivity property holds true for the family
of localization functions $(\eta_1,\ldots,\eta_K)$:
\begin{itemize}[leftmargin=0.7cm]
\item[vii)] Let $m\in\mathcal{K}^{\mathrm{2ph}}$ be a two-phase interface. Then
						\begin{align}\label{CoercivityTwoPhaseInterface}
					  |\dist(\cdot,\mathcal{T}_{m}(t))|^2\wedge 1\leq C(1{-}\eta_m(\cdot,t))
						\quad\text{on}\quad \Rd[2]\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}B_{\tilde r_c}(\mathcal{T}_k(t)).
					  \end{align}
						Let $k\in\mathcal{K}^{\mathrm{3j}}$ be such that the two-phase interface $\mathcal{T}_m\subset {\bar{I}}_{i,j}$
						has an endpoint at the triple junction $\mathcal{T}_k$. Then
						\begin{align}\label{CoercivityWedgeAroundInterface}
					  |\dist(\cdot,\mathcal{T}_{m}(t))|^2\leq C(1{-}\eta_m(\cdot,t){-}\eta_k(\cdot,t))
						\quad\text{on}\quad B_{\tilde r_c}(\mathcal{T}_k(t))\cap W_{i,j}(t).
					  \end{align}
						Let $\mathcal{T}_l\subset I_{i,p},\,l\neq m,$ be another two-phase interface with an endpoint
						at the triple junction $\mathcal{T}_k$. Then
						\begin{align}\label{CoercivityInterpolationWedge}
					  |\dist(\cdot,\mathcal{T}_{k}(t))|^2\leq C(1{-}\eta_m(\cdot,t){-}\eta_k(\cdot,t){-}\eta_l(\cdot,t))
						\quad\text{on}\quad B_{\tilde r_c}(\mathcal{T}_k(t))\cap W_{i}(t).
					  \end{align}
\end{itemize}
\end{lemma}

\begin{proof}
For the definition of a partition of unity $(\eta,\eta_1,\ldots,\eta_K)$
with the required localization and coercivity properties we proceed as follows.
Let $\theta$ be a smooth and even cutoff function with $\theta(r)=1$ for $|r|\leq\frac{1}{2}$ and $\theta\equiv 0$
for $|r|\geq 1$. Let $\zeta^{\mathrm{2ph}},\zeta^{\mathrm{3j}}\colon\Rd[]\to[0,\infty)$ be another two smooth cutoff functions defined by
\begin{align}\label{QuadraticInterfaceCutOff}
\zeta^{\mathrm{2ph}}(r) = (1-r^2)\theta(r)
\end{align}
as well as
\begin{align}\label{QuadraticHalfSpaceCutOff}
\zeta^{\mathrm{3j}}(r) = \zeta^{\mathrm{2ph}}(\min\{r,0\}),
\end{align}

\textit{Step 1: $k\in\mathcal{K}^{\mathrm{3j}}$.}
We first define $\eta_k$ for triple junctions $k\in\mathcal{K}^{\mathrm{3j}}$. To this end,
let us assume that the phases $i,j,p\in\{1,\ldots,P\}$ are present
at the triple junction $\mathcal{T}_k$, and the corresponding interfaces are
denoted by $\mathcal{T}_{l_1}\subset {\bar{I}}_{i,j}$, $\mathcal{T}_{l_2}\subset  {\bar{I}}_{j,p}$
and $\mathcal{T}_{l_3}\subset {\bar{I}}_{p,i}$. Let $0<\widetilde r_c\leq \frac{r_c}{2}$ and $0<c_1,c_2\leq 1$
yet to be determined constants. By choosing $\widetilde r_c\leq \frac{r_c}{2}$ sufficiently small,
we may assume that the interfaces $(\mathcal{T}_{l_1},\mathcal{T}_{l_2},\mathcal{T}_{l_3})$ are
indeed a triod with triple junction $\mathcal{T}_{k}$ evolving by mean curvature in the space-time domain
$\bigcup_{t\in [0,T_{\mathrm{strong}})}B_{2\widetilde r_c}(\mathcal{T}_k(t))\times\{t\}$
in the precise sense of Definition~\ref{def:triod}.

We want to define $\eta_k$ such that $\supp\eta_k\subset B_{\widetilde r_c}(\mathcal{T}_k)$.
Recall from Definition~\ref{def:triod} that $B_{\widetilde r_c}(\mathcal{T}_k)$
decomposes into six wedges. Three of them, namely the wedges $W_{i,j}$, $W_{j,p}$ resp.\ $W_{p,i}$,
contain the interfaces $\mathcal{T}_{l_1}$, $\mathcal{T}_{l_2}$ resp.\ $\mathcal{T}_{l_3}$.
The other three are interpolation wedges denoted by $W_{i}$, $W_{j}$ resp.\ $W_{p}$.
Finally, recall that we identified the set $W_{i}\cup W_{j}\cup W_{i,j}$ as the intersection
of certain halfspaces $\mathbb{H}_{i}\cap\mathbb{H}_{j}$.
We then define an auxiliary cutoff
\begin{align}\label{DefinitionAuxiliaryCutoffHalfSpace}
\zeta^{\mathrm{3j}}_{i,j}(x,t) := \zeta^{\mathrm{3j}}\Big(\frac{\sigdist(x,\mathbb{H}_{i}(t))}{c_2\widetilde r_c}\Big)
\zeta^{\mathrm{3j}}\Big(\frac{\sigdist(x,\mathbb{H}_{j}(t))}{c_2\widetilde r_c}\Big),\quad
(x,t)\in \Rd[2]\times [0,T_{\mathrm{strong}}),
\end{align}
and analogously the cutoffs $\zeta^{\mathrm{3j}}_{j,p}$ resp.\ $\zeta^{\mathrm{3j}}_{p,i}$.
Recall that the convention is that $\sigdist(\cdot,\mathbb{H}_i)$ is negative
in the half-space $\mathbb{H}_i$ and positive outside. Moreover,
let us introduce the auxiliary cutoff function
\begin{align}\label{DefinitionAuxiliaryCutoffInterface}
\zeta^{\mathrm{2ph}}_{i,j}(x,t) := \zeta^{\mathrm{2ph}}\Big(
\frac{\sigdist(x,{\bar{I}}_{i,j}(t))}{c_1\widetilde r_c}\Big),\quad
(x,t)\in \Rd[2]\times [0,T_{\mathrm{strong}}),
\end{align}
with an analogous definition for the cutoffs $\zeta^{\mathrm{2ph}}_{j,p}$ resp.\ $\zeta^{\mathrm{2ph}}_{p,i}$.
Hence, $\zeta^{\mathrm{3j}}_{i,j}$ is a cut-off for the wedge $\mathbb{H}_i\cap\mathbb{H}_j$.
By now, we have everything in place to move on with the definition of $\eta_k$.
We begin by setting
\begin{align}\label{DefinitionEtaTripleJunctionWedgeInterface}
\eta_k(x,t) := \zeta^{\mathrm{3j}}_{i,j}(x,t)\zeta^{\mathrm{2ph}}_{i,j}(x,t),\quad
t\in [0,T_{\mathrm{strong}}),\,x\in B_{\widetilde r_c}(\mathcal{T}_k(t))\cap W_{i,j}(t),
\end{align}
and analogously on the other wedges $W_{j,p}$ and $W_{p,i}$. To define $\eta_k$ on
the interpolation wedges, we make use of the interpolation parameter
as built in Lemma~\ref{lemma:interpolation_functions}. To clarify the direction
of interpolation, i.e., on which boundary of the interpolation wedge the corresponding
interpolation function is equal to one or zero, we make use of the following
notational convention. For a triple junction $\mathcal{T}_k$ with the
phases $i,j,p\in\{1,\ldots,P\}$ being present let us define two interpolation functions for
each interpolation wedge. For the interpolation wedge $W_i$, say, we denote
by $\lambda_{i}^{j}$ the interpolation function as built in Lemma~\ref{lemma:interpolation_functions}
and which is equal to one on $(\partial W_{i,j}\cap\partial W_{i})\setminus\mathcal{T}_{k}$ and
which vanishes on $(\partial W_{p,i}\cap\partial W_{i})\setminus\mathcal{T}_{k}$. We also define
$\lambda_{i}^{p}:= 1-\lambda_{i}^{j}$ which interpolates on $W_{i}$ in the opposite direction.
Analogously, one introduces the interpolation functions on the other interpolation wedges.
Finally, we may then define
\begin{equation}\label{DefinitionEtaTripleJunctionInterpolationWedge}
\begin{aligned}
\eta_k(x,t) &:= \lambda_{i}^{j}(x,t)\zeta^{\mathrm{3j}}_{i,j}(x,t)\zeta^{\mathrm{2ph}}_{i,j}(x,t)
+\lambda_{i}^{p}(x,t)\zeta^{\mathrm{3j}}_{p,i}(x,t)\zeta^{\mathrm{2ph}}_{p,i}(x,t),
\\&~~~~~
t\in [0,T_{\mathrm{strong}}),\,x\in B_{\widetilde r_c}(\mathcal{T}_k(t))\cap W_{i}(t),
\end{aligned}
\end{equation}
with an analogous definition on the other two interpolation wedges $W_{j}$ and $W_{p}$.
Outside of $B_{\widetilde r_c}(\mathcal{T}_k)$, we of course simply set $\eta_k\equiv 0$.
We refer to Figure~\ref{fig:cutoff} for an illustration of the construction.

Choosing $c_1$ and $c_2$ small enough (depending only on the surface tension matrix $\sigma\in\Rd[P\times P]$),
we indeed obtain \eqref{LocalizationTripleJunction}. The required regularity for $\eta_k$ follows
from the regularity of the strong solution, see Definition~\ref{DefinitionStrongSolution},
the fact that $\widetilde r_c\leq \frac{r_c}{2}$, the property \eqref{LocalizationTripleJunction}
as well as the controlled blowup of the derivatives of the interpolation parameter
at the triple junction \eqref{boundslambda1} and \eqref{boundslambda2}. Finally, it is just a
matter of possibly choosing $\widetilde r_c\leq \frac{r_c}{2}$
even smaller, such that also \eqref{LocalizationTwoTripleJunctions} holds true.

\textit{Step 2: $m\in\mathcal{K}^{\mathrm{2ph}}$.}
We next define $\eta_m$ for a smooth two-phase interface $\mathcal{T}_m\subset {\bar{I}}_{i,j}$.
That $\mathcal{T}_m$ is a smoothly evolving interface
by mean curvature in the space-time tubular neighborhood
$\bigcup_{t\in[0,T_{\mathrm{strong}})}\big(\mathcal{T}_{m}(t){+}B_{2\widetilde r_c}(0)\big)\times\{t\}$
in the precise sense of Definition~\ref{DefinitionStrongSolutionTwoPhase}
is immediate from the definition of a strong solution, see Definition~\ref{DefinitionStrongSolution}.

If the interface $\mathcal{T}_m\subset {\bar{I}}_{i,j}$ has no endpoint at a triple junction, we simply set
\begin{align}\label{DefinitionEtaTwoPhaseNoTripleJunction}
\eta_m(x,t) := \zeta^{\mathrm{2ph}}_{i,j}(x,t),\quad
(x,t)\in \Rd[2]\times[0,T_{\mathrm{strong}}),
\end{align}
where the cutoff $\zeta^{\mathrm{2ph}}_{i,j}$ was already defined in \eqref{DefinitionAuxiliaryCutoffInterface}.
If the interface $\mathcal{T}_m\subset {\bar{I}}_{i,j}$ has an endpoint at exactly one triple junction,
say $\mathcal{T}_k$ with phases $i,j$ and $p$ being present, we proceed as follows.
Away from the triple junction $\mathcal{T}_k$, we still define
\begin{align}\label{DefinitionEtaTwoPhaseAwayTripleJunction}
\eta_m(x,t) := \zeta^{\mathrm{2ph}}_{i,j}(x,t),\quad
t\in[0,T_{\mathrm{strong}}),\,x\in \Rd[2]\setminus B_{\widetilde r_c}(\mathcal{T}_k(t)).
\end{align}
Near the triple junction, i.e., on $B_{\widetilde r_c}(\mathcal{T}_k)$, we aim to modify the definition
such that $\eta_m$ is supported within the set $W_{i}\cup W_{j}\cup W_{i,j}$.
To this end, we define
\begin{align}\label{DefinitionEtaTwoPhaseWedgeInterface}
\eta_m(x,t) := (1{-}\zeta^{\mathrm{3j}}_{i,j}(x,t))\zeta^{\mathrm{2ph}}_{i,j}(x,t),\quad
x\in B_{\widetilde r_c}(\mathcal{T}_k(t))\cap W_{i,j}(t),
\end{align}
where the auxiliary cutoff $\zeta^{\mathrm{3j}}_{i,j}$ was introduced in \eqref{DefinitionAuxiliaryCutoffHalfSpace}.
On the interpolation wedges $W_i$ resp.\ $W_j$, we again make use of the
interpolation parameter and set
\begin{equation}\label{DefinitionEtaTwoPhaseInterpolationWedge}
\begin{aligned}
\eta_m(x,t) &:= \lambda_{i}^{j}(x,t)(1{-}\zeta^{\mathrm{3j}}_{i,j}(x,t))\zeta^{\mathrm{2ph}}_{i,j}(x,t),\quad
x\in B_{\widetilde r_c}(\mathcal{T}_k(t))\cap W_{i}(t) \\
\eta_m(x,t) &:= \lambda_{j}^{i}(x,t)(1{-}\zeta^{\mathrm{3j}}_{i,j}(x,t))\zeta^{\mathrm{2ph}}_{i,j}(x,t),\quad
x\in B_{\widetilde r_c}(\mathcal{T}_k(t))\cap W_{j}(t).
\end{aligned}
\end{equation}
We refer again to Figure~\ref{fig:cutoff} for an illustration of the construction.
Finally, if the endpoints of the interface $\mathcal{T}_m\subset {\bar{I}}_{i,j}$ are
located at two distinct triple junctions, we simply repeat the preceding procedure
at the second triple junction.

The required regularity for $\eta_m$ follows for the same reasons as for
the localization functions $\eta_k$ around triple junctions $k\in\mathcal{K}^{\mathrm{3j}}$.
Property \eqref{LocalizationTwoPhase} is a consequence of choosing
$c_1$ and $c_2$ small enough (depending only on the surface tension matrix
$\sigma\in\Rd[P\times P]$). The statements preceding the relations
\eqref{LocalizationTripleJunctionTwoPhase} resp.\ \eqref{LocalizationTwoInterfaces}
follow by possibly reducing $\widetilde r_c\leq \frac{r_c}{2}$ even further.
Finally, the precise localization properties
\eqref{LocalizationTripleJunctionTwoPhase} and \eqref{LocalizationTwoInterfaces}
are then immediate consequences of our definitions \eqref{DefinitionEtaTwoPhaseWedgeInterface}
and \eqref{DefinitionEtaTwoPhaseInterpolationWedge}.

\textit{Step 3: Partition of Unity.}
Next, we validate the partition of unity property
for the family of localization functions $(\eta_1,\ldots,\eta_K)$.
First of all, it is clear from our definitions
\eqref{DefinitionEtaTripleJunctionWedgeInterface}--\eqref{DefinitionEtaTwoPhaseInterpolationWedge}
that $\eta_k\in [0,1]$ for each topological feature $k\in\{1,\ldots,K\}$.
Together with the already established localization properties
\eqref{LocalizationTwoPhase}--\eqref{LocalizationTwoInterfaces} and the definitions
\eqref{DefinitionEtaTripleJunctionWedgeInterface}--\eqref{DefinitionEtaTwoPhaseInterpolationWedge},
it also follows that $\sum_{k=1}^K\eta_k\leq 1$ on $\Rd[2]\times [0,T_{\mathrm{strong}})$
as well as $\sum_{k=1}^K\eta_k\equiv 1$ on the evolving network of interfaces $({\bar{I}}_{i,j}(t))_{i\neq j}$.
Hence, we may define the bulk term $\eta:=1-\sum_{k=1}^K\eta_k\in [0,1]$ and obtain that the
extended family $(\eta,\eta_1,\ldots,\eta_K)$ is indeed a partition of unity on $\Rd[2]\times [0,T_{\mathrm{strong}})$.

\textit{Step 4: Coercivity estimates.}
The bound \eqref{CoercivityTwoPhaseInterface} is a consequence of the definition
of the quadratic cutoff \eqref{QuadraticInterfaceCutOff}
as well as the definitions \eqref{DefinitionEtaTwoPhaseNoTripleJunction} resp.\ \eqref{DefinitionEtaTwoPhaseAwayTripleJunction}.
The estimate \eqref{CoercivityWedgeAroundInterface} follows from \eqref{DefinitionEtaTwoPhaseWedgeInterface},
\eqref{DefinitionEtaTripleJunctionWedgeInterface} and again the definition \eqref{QuadraticInterfaceCutOff}.
Finally, the bound \eqref{CoercivityInterpolationWedge} can be inferred from
\eqref{DefinitionEtaTripleJunctionInterpolationWedge}, \eqref{DefinitionEtaTwoPhaseInterpolationWedge},
the definition \eqref{QuadraticInterfaceCutOff} as well as the estimate
\begin{align*}
|\dist(x,\mathcal{T}_k)|^2
\leq
C\lambda_{i}^{j}(x)|\dist(x,{\bar{I}}_{i,j})|^2
+ C\lambda_{i}^{p}(x)|\dist(x,{\bar{I}}_{p,i})|^2,
\end{align*}
which holds on an interpolation wedge $W_{i}$ at a triple junction $\mathcal{T}_k$ with
phases $i,j$ and $p$ being present. This concludes the proof of Lemma~\ref{LemmaPartitionOfUnity}.
\end{proof}

At this point, we have everything in place to move on with the construction of the global vector
fields $\xi_{i,j}$ and the global velocity field $B$. Once we defined the latter, we may also
prove that the localization functions $(\eta_1,\ldots,\eta_K)$ are subject to an advection
equation up to controlled error terms.

\begin{figure}
\includegraphics[scale=0.12]{cutoff_2phase}
\,
\includegraphics[scale=0.12]{cutoff_3junc}
\\
\includegraphics[scale=0.12]{cutoff_all2phase}
\,
\includegraphics[scale=0.12]{cutoff_partitionofunity}
\caption{The different functions $\eta_k$ in the partition of unity at a single triple junction: The function $\eta_k$ for a single two-phase interface $\mathcal{T}_k={\bar{I}}_{i,j}$ ending at the triple junction (top left), the function $\eta_k$ for the triple junction $\mathcal{T}_k=\{p_k\}$ itself (top right), the sum of all two-phase localization functions at a triple junction (bottom left), and the sum of all localization functions $\sum_k \eta_k=1-\eta$ (bottom right). Observe that the sum of all localization functions equals $1$ on the interfaces in the strong solution, but decays quadratically away from them.}
\label{fig:cutoff}
\end{figure}

\subsection{Global construction of the calibration}\label{SectionGlobalDefinitions}
This section is devoted to the construction of the vector fields $\xi_{i,j}$, extending
the unit-normal vector fields of the network of interfaces of a strong solution, as well
as the velocity field $B$ along which all constructions are approximately transported
during the time evolution of the network of interfaces.
Given a strong solution $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution}, recall
that we decomposed the associated network of interfaces into $K$ topological features $\mathcal{T}_1,\ldots,\mathcal{T}_K$.
More precisely, we defined $\{1,\ldots,K\}=:\mathcal{K}^{\mathrm{2ph}}\cup\mathcal{K}^{\mathrm{3j}}$, with $\mathcal{K}^{\mathrm{3j}}$ enumerating the triple junctions $p_{k}$ present
in the strong solution, whereas $\mathcal{K}^{\mathrm{2ph}}$ enumerates the space-time connected component of the
smooth two-phase interfaces ${\bar{I}}_{i,j}$.
We then put $\mathcal{T}_k:=\{p_{k}\}$, if $k\in\mathcal{K}^{\mathrm{3j}}$, or $\mathcal{T}_k\subset {\bar{I}}_{i,j}$
for the corresponding space-time connected component $k\in\mathcal{K}^{\mathrm{2ph}}$ of a two-phase interface ${\bar{I}}_{i,j}$.

The idea for the construction of the vector fields $\xi_{i,j}$ is as follows.
First, we provide the definition of local vector fields $\xi^k_{i,j}$ in the support of the associated
localization function $\eta_k$ for each topological feature $\mathcal{T}_k$. If both the phases $i$ and $j$
are \emph{present} at $\mathcal{T}_k$,
we define $\xi^k_{i,j}$ by means of the local constructions provided in Section~\ref{SectionLocalConstructionsTwoPhase}
for the model problem of a smooth manifold resp.\ Section~\ref{SectionLocalConstructionsTriod} for the model problem
of a triod. This, however, leaves open the question of the definition of the vector fields $\xi_{i,j}^k$
for phases \emph{absent} at $\mathcal{T}_k$. It turns out that the question of the proper definition of the vector fields
for phases being absent at $\mathcal{T}_k$ is associated with the conditions of global stability between the phases.
In particular, we would like to ensure that at a given topological feature $\mathcal{T}_{k}$,
our relative entropy functional provides a length control for those interfaces not being present at $\mathcal{T}_k$.
For this purpose, we rely on the stability conditions provided by \eqref{TriangleInequalitySurfaceTensions} and
\eqref{NoNucleationSurfaceTensions} for an admissible matrix of surface tension in the sense
of Definition~\ref{DefinitionAdmissibleSurfaceTensions}.

\begin{lemma}\label{LemmaLocalConstructionsNetwork}
Let $\sigma\in\Rd[P\times P]$ be an admissible matrix of surface tensions
in the sense of Definition~\ref{DefinitionAdmissibleSurfaceTensions}, and
let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution}.
Let $(\eta,\eta_1,\ldots,\eta_K)$ be a partition of unity as constructed in
Lemma~\ref{LemmaPartitionOfUnity}. Then for every choice of distinct phases $i,j\in\{1,\ldots,P\}$
and every topological feature $k\in\{1,\ldots,K\}$ there exist vector fields
\begin{align*}
\xi^k_{i,j}\colon\bigcup_{t\in [0,T_{\mathrm{strong}})}\supp\eta_k(\cdot,t)\times\{t\}&\mapsto\Rd[2], \\
\xi^k_{i}\colon\bigcup_{t\in [0,T_{\mathrm{strong}})}\supp\eta_k(\cdot,t)\times\{t\}&\mapsto\Rd[2],
\end{align*}
where the former are defined by means of the local vector fields constructed in
Section~\ref{SectionLocalConstructionsTwoPhase} resp.\ Section~\ref{SectionLocalConstructionsTriod},
satisfying the following properties:
\begin{itemize}[leftmargin=0.7cm]
\item[i)] It holds $\xi^k_{i,j},\xi^k_{i}\in L^\infty_tW^{2,\infty}_x\cap W^{1,\infty}_tC^0_x$.
\item[ii)] We have $\xi^k_{i,j}=-\xi^k_{j,i},\,|\xi^k_{i,j}|\leq 1$ as well as
					 \begin{align}\label{CalibrationLocalConstruction}
					 \sigma_{i,j}\xi^k_{i,j}=\xi_i^k-\xi_j^k.
					 \end{align}
\item[iii)] If the phases $i$ and $j$ are both present at the topological feature $\mathcal{T}_k$,
						then $\xi^k_{i,j}$ coincides with $\bar{\vec{n}}_{i,j}$ on $\supp\eta_k\cap {\bar{I}}_{i,j}$.
\item[iv)] There exists a constant $c=c(\sigma)\in (0,1)$ with the property, that
					 if either phase $i$ or $j$ is absent at the topological feature $\mathcal{T}_k$, then
					 \begin{align}\label{CoercivityMinorityPhase}
					 |\xi^k_{i,j}|\leq c<1.
					 \end{align}
\end{itemize}
\end{lemma}

\begin{proof}
The proof consists of two parts distinguishing between the topological
features present in the network of interfaces of the strong solution.

\textit{Step 1: Two-phase interface.}
Let $i,j\in\{1,\ldots,P\}$ with $i\neq j$ be fixed, and let $k\in\mathcal{K}^{\mathrm{2ph}}$. We
first assume that both the phases $i$ and $j$ are present at the two-phase
interface $\mathcal{T}_{k}$, i.e., $\mathcal{T}_k\subset {\bar{I}}_{i,j}$. We then define the
vector field $\smash{\xi_{i,j}^k}$ in the support of $\eta_k$ as in Lemma~\ref{LemmaBoundsLocalConstructionsTwoPhase}.
Note that by the localization property \textit{ii)} from Lemma~\ref{LemmaPartitionOfUnity}
of the partition of unity $(\eta_1,\ldots,\eta_K)$,
we are indeed in the setting of Section~\ref{SectionLocalConstructionsTwoPhase}.
In particular, $\xi^k_{i,j}=-\xi^k_{j,i}$ and $\xi^k_{i,j}$ coincides with $\bar{\vec{n}}_{i,j}$
on $\supp\eta_k\cap {\bar{I}}_{i,j}$.
Furthermore, let us define the vector fields $\xi^k_{i}$ resp.\ $\xi^k_{j}$
as $\xi_i^k:=\smash{\frac{\sigma_{i,j}}{2} \xi_{i,j}^k}$ resp.\ as
$\xi_j^k:=\smash{\frac{\sigma_{i,j}}{2} \xi_{j,i}^k}$. This ensures that the
desired formula \eqref{CalibrationLocalConstruction} is indeed satisfied.

Now, let us assume that either phase $i$ or $j$ is absent at the
two-phase interface $\mathcal{T}_{k}$. To be specific, we fix $m,l\in\{1,\ldots,P\}$
with $m\neq l$ such that $\mathcal{T}_{k}\subset {\bar{I}}_{m,l}$. The idea now is to
first define vector fields $\xi^k_{i}$ resp.\ $\xi^k_{j}$ and then define $\xi^k_{i,j}$
by means of \eqref{CalibrationLocalConstruction} such that \eqref{CoercivityMinorityPhase}
holds true. To this end, we rely on the strict triangle inequality \eqref{TriangleInequalitySurfaceTensions}
for the given matrix of surface tensions. Let us define
\begin{align*}
\xi^k_i &:= \frac{1}{2}(\sigma_{l,i}\xi^k_{m,l}+\sigma_{m,i}\xi^k_{l,m}),
\end{align*}
and analogously for $\xi^k_{j}$. Note that this is indeed well-defined since we already provided a definition
for the vector fields $\xi^k_{m,l}=-\xi^k_{l,m}$ associated to the phases present at $\mathcal{T}_k$.
This definition is also consistent with the previous one because of the convention $\sigma_{i,i}=0$.
We may then compute plugging in the definitions
\begin{align*}
\xi^k_{i,j}:=\frac{\xi^k_i-\xi^k_j}{\sigma_{i,j}} =
\frac{1}{2}\Big(\frac{\sigma_{l,i}-\sigma_{l,j}}{\sigma_{i,j}}\xi^k_{m,l}
+\frac{\sigma_{m,i}-\sigma_{m,j}}{\sigma_{i,j}}\xi^k_{l,m}\Big).
\end{align*}
Hence, \eqref{CoercivityMinorityPhase} holds true because we have
$|\frac{\sigma_{l,i}-\sigma_{l,j}}{\sigma_{i,j}}|<1$ and
$|\frac{\sigma_{m,i}-\sigma_{m,j}}{\sigma_{i,j}}|<1$ due to
the strict triangle inequality \eqref{TriangleInequalitySurfaceTensions}.

\textit{Step 2: Triple junctions.}
Let now $k\in\mathcal{K}^{\mathrm{3j}}$. Again, we first assume that both the phases $i$ and $j$ are present
at the triple junction $\mathcal{T}_{k}$, i.e., a connected component of the
interface ${\bar{I}}_{i,j}$ has an endpoint at $\mathcal{T}_{k}$. Note that by the
localization property \textit{iii)} from Lemma~\ref{LemmaPartitionOfUnity}
of the partition of unity $(\eta_1,\ldots,\eta_K)$,
we are in the setting of Section~\ref{SectionLocalConstructionsTriod}.
We then define the vector field $\xi^k_{i,j}$ in the support of $\eta_k$
by the corresponding vector field from Proposition~\ref{prop:xi_triple_junction}.
In particular, $\xi^k_{i,j}=-\xi^k_{j,i}$ and $\xi^k_{i,j}$ coincides with $\bar{\vec{n}}_{i,j}$
on $\supp\eta_k\cap {\bar{I}}_{i,j}$.

Assume now that $p\in\{1,\ldots,P\}$ is the third phase being present
at the triple junction $\mathcal{T}_{k}$. By construction, we have
$\sigma_{i,j}\xi^{\,k}_{i,j}+\sigma_{j,p}\xi^{\,k}_{j,p}+\sigma_{p,i}\xi^{\,k}_{p,i}=0$
on the support of $\eta_k$. Defining then the vector field $\xi^k_{i}$
as $\xi^k_{i}:=\frac{1}{3}(\sigma_{i,j}\xi^k_{i,j}+\sigma_{i,p}\xi^k_{i,p})$,
and analogously for $\xi^k_{j}$ resp.\ $\xi^k_{p}$, we indeed obtain \eqref{CalibrationLocalConstruction}.

Finally, let us assume that either phase $i$ or $j$ is absent at the triple junction $\mathcal{T}_k$.
To be specific, we assume that the phases $l,m,n\in\{1,\ldots,P\}$ are present at $\mathcal{T}_k$.
We then employ the stability condition for triple junctions \eqref{NoNucleationSurfaceTensions}
to construct the vector fields $\smash{\xi_{i,j}^k}$. This condition allows us to interpret the matrix
of surface tensions $\smash{\sigma=(\sigma_{ij})_{i,j=1,\dots,P}}$ as pairwise (Euclidean) distances
of points in $\mathbb{R}^P$. (Recall that $\sigma_{i,i}=0$ by convention.)
That means we can find points $\smash{q_1,\dots,q_P\in \mathbb{R}^{P-1}}$ such that
\begin{align*}
\sigma_{i,j} = |q_i-q_j| \quad \text{for all } i,j=1,\dots,P,
\end{align*}
see \cite[Theorem 1]{Schoenberg35}.
Note that with $Q$, cf.~\eqref{NoNucleationSurfaceTensions}, also its $3$-minor corresponding to
the indices $l,m,n$ has full rank, so the triangle $q_lq_mq_n$ is non-degenerate and spans a plane
$E=E_{l,m,n}$ in $\Rd[P-1]$. We momentarily define the orthogonal projection onto $E$ by $\pi=\pi_{l,m,n}$.

Now we first construct vectors which define the missing vector fields consistently \textit{at the triple point} itself.
Then their extension to the whole patch around the triple junction will be straightforward.
At the triple point, we set
\begin{align*}
	\xi_i^k := R^k\pi_{l,m,n} q_i,\quad \text{for all } i=1,\dots,P,
\end{align*}
where the orthogonal matrix $R^k\in \mathcal{O}(2)$ is still to be determined (to ensure
consistency with the already provided definitions for the phases present at $\mathcal{T}_k$).
In particular, $\smash{\xi_i^k = R^kq_i}$ for the phases $l,m,n$. Then we define
$\smash{\xi_{i,j}^k}$ in terms of the vectors $\xi_i^k$ as before in the two-phase case via
formula \eqref{CalibrationLocalConstruction}.

\begin{figure}
 	\begin{tikzpicture}[scale=5]
			\tikzmath{\l = .6;};
			\coordinate (A) at (0,0);
			\coordinate (B) at (30:\l);
			\draw [name path=A--B,color=white] (A) -- (B);
			\coordinate (C) at (0:\l);
			\coordinate (D) at ($(0:\l)+(130:\l)$);
			\draw [name path=C--D,color=white] (C) -- (D);
			\coordinate (E) at (intersection of A--B and C--D);

		  	\draw (0,0) -- node[label= below:{$\sigma_{l,m}$}] {} (0:\l);
		  	\draw (0:\l) --  node[label= right:{$\sigma_{m,n}$}] {}  (E);
		  	\draw (E) --node[label= left:{$\sigma_{n,l}$}] {}  (0,0);

		  	\filldraw (0,0) circle[radius=.25pt];
			\node[below, outer sep=2pt] (0,0) {$q_l$};
			\filldraw (\l,0) circle[radius=.25pt];
			\node[below, outer sep=2pt] at (\l,0) {$q_m$};
			\filldraw (E) circle[radius=.25pt];
			\node[above, outer sep=2pt] at (E) {$q_n$};

			\filldraw (.05,.4) circle[radius=.25pt];
			\node[above, outer sep=2pt] at (.05,.4) {$q_i$};
			\draw[dashed] (E) --node[label= below:{$<\sigma_{i,l}$}] {} (.05,.4);
			\filldraw (.8,.4) circle[radius=.25pt];
			\node[above, outer sep=2pt] at (.8,.4) {$q_j$};
			\draw[dashed] (E) --node[label= below:{$<\sigma_{j,l}$}] {} (.8,.4);
			\draw[dashed] (.05,.4) --node[label= above:{$<\sigma_{i,j}$}] {} (.8,.4);
		\end{tikzpicture}
		\caption{Sketch of the $l^2$-embedding of $\sigma$ in the case that $i$ and $j$ correspond to absent phases, projected into the plane containing $q_l$, $q_m$ and $q_n$. \label{fig:embedding}}
\end{figure}

Note that after projecting an out-of-plane vector onto $E$ via $\pi=\pi_{l,m,n}$ it becomes short. Hence, we have
\begin{align}
\label{eq:short}
	\begin{cases}
		|\xi_{i,j}^k|=1 &\text{if } i,j \in \{l,m,n\}\\
		|\xi_{i,j}^k|\leq c < 1&  \text{otherwise.}
	\end{cases}
\end{align}
Indeed, if $i,j\in \{l,m,n\}$, both $q_i$ and $q_j$ lie in $E$ and hence the projection leaves them invariant.
We handle the second case in two subcases:

If exactly one of the two indices, say, $j$ corresponds to a phase being present at $\mathcal{T}_k$, then $\pi q_j = q_j$.
Note that with $Q$, also each $4$-minor has full rank, so that $q_i$ cannot lie in the plane $E$,
hence $\pi q_i \neq q_i$, which implies the strict inequality in this subcase.
If both $i$ and $j$ correspond to phases being absent at $\mathcal{T}_k$, we only need to argue that $q_i-q_j$ does not
lie in the linear space $E-q_l$. Otherwise the $5$-minor of $Q$ corresponding to the indices $l,m,n,i,j$
were not of full rank, a contradiction. So indeed, \eqref{eq:short} holds.

Thanks to the compatibility condition
\begin{align*}
\sigma_{l,m} \xi_{l,m}^k + \sigma_{m,n}\xi_{m,n}^k + \sigma_{n,l}\xi_{n,l}^k =0
\end{align*}
we can find $R^k$ such that $\smash{\xi_{l,m}^k= \vec{\bar n}_{l,m}}$, $\smash{\xi_{m,n}^k= \vec{\bar n}_{m,n}}$
as well as $\smash{\xi_{n,l}^k= \vec{\bar n}_{n,l}}$ at the triple junction.
In particular, the construction is consistent with the previous constructions
for the phases present at $\mathcal{T}_k$. Note also that $R^k$ is indeed orthogonal
since $\smash{|\xi^k_{l,m}|=|\xi^k_{m,n}|=|\xi^k_{n,l}|=1}$.

To extend the definition of $\smash{\xi_{i,j}^k}$ (resp.\ $\xi^k_{i}$) to the whole support of $\eta_k$,
we first express $\smash{\xi_{i,j}^k}$ (resp.\ $\xi^k_{i}$) \textit{at the triple junction}
as a fixed linear combination of the vector fields of the phases present at the triple junction:
there exist time-independent coefficients $\lambda,\lambda',\lambda''$ such that
\begin{align*}
\xi^k_{i} = \lambda\xi^k_l + \lambda'\xi^k_m + \lambda''\xi^k_n.
\end{align*}
Since the vector fields on the right hand side are already defined on the whole support of $\eta_k$,
we may extend the definition of the vector fields $\xi^k_i$ for the phases absent at the triple
junction $\mathcal{T}_k$ by means of this fixed linear combination, and then, as already said,
define the missing vector fields $\xi^k_{i,j}$ via \eqref{CalibrationLocalConstruction}.
In particular, the desired properties \eqref{CalibrationLocalConstruction}
as well as \eqref{CoercivityMinorityPhase} continue to hold true. This concludes the proof of
Lemma~\ref{LemmaLocalConstructionsNetwork}.
\end{proof}

\begin{figure}
\includegraphics[scale=0.15]{lengthXi}
\caption{Plot of the length of the vector field $\xi_{i,j}$. Observe that the length is $1$ on the interface ${\bar{I}}_{i,j}$ of the strong solution, but decays quadratically away from it to a value strictly smaller than $1$, even on the other interfaces ${\bar{I}}_{i,p}$ and ${\bar{I}}_{j,p}$. As a consequence, the integral $\int_{I_{i,j}} 1-\vec{n}_{i,j}\cdot \xi_{i,j} \,d\mathcal{H}^{1}$ provides an upper bound for the interface error functional $c\smash{\int_{I_{i,j}} \min\{\dist^2(x,{\bar{I}}_{i,j}),\,1\} \,d\mathcal{H}^{1}}$.}
\label{fig:lengthXi}
\end{figure}

Now we may define the global extensions $\xi_{ij}=-\xi_{ji}$ of the unit normal
vector fields between the phases $i$ and $j$ in the strong solution by gluing the local definitions by
means of the partition of unity $(\eta,\eta_1,\ldots,\eta_K)$ from Lemma~\ref{LemmaPartitionOfUnity}.

\begin{construction}
\label{DefinitionGlobalXi}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution} on some time interval $[0,T_{\mathrm{strong}})$.
Let $(\eta,\eta_1,\ldots,\eta_K)$ be a partition of unity as constructed in
Lemma~\ref{LemmaPartitionOfUnity}. Let the local vector fields $\xi^k_{i,j}=-\xi^k_{j,i}$
be given as in Lemma~\ref{LemmaLocalConstructionsNetwork}.
For all $i,j\in\{1,\ldots,P\}$ with $i\neq j$, we then define
\begin{align}\label{GlobalXi}
\xi_{i,j}(x,t) &:= \sum_{k=1}^K \eta_k(x,t) \xi_{i,j}^k(x,t)
\end{align}
for all $x\in \Rd[2]$ and all $t\in [0,T_{\mathrm{strong}})$.
\end{construction}

We proceed with the derivation of the coercivity condition provided by
the length of the vector fields $\xi_{i,j}$ as defined by
Construction~\ref{DefinitionGlobalXi}. For an illustration we
refer to Figure~\ref{fig:lengthXi}.

\begin{lemma}
\label{lem:coercivityLenghtXi}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution} on some time interval $[0,T_{\mathrm{strong}})$.
Let $(\eta,\eta_1,\ldots,\eta_K)$ be a partition of unity as constructed in
Lemma~\ref{LemmaPartitionOfUnity}, and let $\xi_{i,j}$ be the family of vector
fields provided by Construction~\ref{DefinitionGlobalXi}. Then there exists $C=C(\bar\chi)>0$
such that for all $i,j\in\{1,\ldots,P\}$ with $i\neq j$ it holds
\begin{align}
\label{eq:coercivityLengthXi}
\dist^2(\cdot,{\bar{I}}_{i,j}(t)) \wedge 1 \leq C(1-|\xi_{i,j}(\cdot,t)|).
\end{align}
\end{lemma}

\begin{proof}
Fix $i,j\in\{1,\ldots,P\}$ with $i\neq j$. The asserted estimate \eqref{eq:coercivityLengthXi}
is trivially fulfilled for $x\notin\supp\xi_{i,j}$. By the definition \eqref{GlobalXi}
we may therefore assume that there exists a topological feature $k\in\{1,\ldots,K\}$ such that
$x\in\supp\eta_k$.

If either phase $i$ or phase $j$ is absent at the topological feature
$\mathcal{T}_k$, we argue based on \eqref{CoercivityMinorityPhase} as follows.
Writing $\xi_{i,j}=\eta_k\xi^k_{i,j} + \sum_{l\in\{1,\ldots,K\}\setminus\{l\}}\eta_l\xi^l_{i,j}$
we deduce from \eqref{CoercivityMinorityPhase} together with the triangle inequality that
there exists $c=c(\bar\chi)\in (0,1)$ such that
$|\xi_{i,j}| \leq 1 - c$. This in turn immediately implies the estimate \eqref{eq:coercivityLengthXi}.

Hence, assume that both the phases $i$ and $j$ are present at $\mathcal{T}_k$. The proof
of \eqref{eq:coercivityLengthXi} in this case essentially boils down to a direct combination
of the localization properties \eqref{LocalizationTwoPhase}--\eqref{LocalizationTwoInterfaces}
and the coercivity estimates \eqref{CoercivityTwoPhaseInterface}--\eqref{CoercivityInterpolationWedge}
of the partition of unity $(\eta,\eta_1,\ldots,\eta_K)$. The precise argument proceeds as follows.

We consider the case of a two-phase interface $k\in\mathcal{K}^{\mathrm{2ph}}$. As it will be clear from
the subsequent argument, the case of a triple junction $k\in\mathcal{K}^{\mathrm{3j}}$ follows by symmetry.
If $x\in\supp\eta_k\setminus\bigcup_{l\in\mathcal{K}^{\mathrm{3j}}}\supp\eta_l$ we may infer from \eqref{CoercivityTwoPhaseInterface}
together with \eqref{LocalizationTripleJunction}, \eqref{LocalizationTripleJunctionTwoPhase}
and \eqref{LocalizationTwoInterfaces} that $\dist^2(x,{\bar{I}}_{i,j})\leq C(1{-}\eta_k(x))
\leq C(1{-}|\xi_{i,j}(x)|)$ as required. Otherwise, we assume that there exists $l\in\mathcal{K}^{\mathrm{3j}}$
such that $x\in\supp\eta_k\cap\supp\eta_l$ (which because of \eqref{LocalizationTwoTripleJunctions}
is the only such triple junction); in other words, the interface $\mathcal{T}_k\subset {\bar{I}}_{i,j}$ has
an endpoint at the triple junction $\mathcal{T}_l$. As a consequence of \eqref{LocalizationTripleJunctionTwoPhase},
it then suffices to verify \eqref{eq:coercivityLengthXi} on the wedge $W_{i,j}$ as well
as its two adjacent interpolation wedges $W_i$ resp.\ $W_j$. In the case of the wedge $W_{i,j}$,
it follows from \eqref{CoercivityWedgeAroundInterface} together with \eqref{LocalizationTripleJunctionTwoPhase}
and \eqref{LocalizationTwoInterfaces} that $\dist^2(x,{\bar{I}}_{i,j})\leq C(1{-}\eta_k(x){-}\eta_m(x))
\leq C(1{-}|\xi_{i,j}(x)|)$. In the remaining case of one of the interpolation wedges,
say $W_i$, we deduce first from \eqref{CoercivityInterpolationWedge} and then followed by
\eqref{LocalizationTripleJunctionTwoPhase} and \eqref{LocalizationTwoInterfaces} that
$\dist^2(x,{\bar{I}}_{i,j})\leq C(1{-}\eta_k(x){-}\eta_m(x)-\eta_l(x))
\leq C(1{-}|\xi_{i,j}(x)|)$. Here, $l\in\mathcal{K}^{\mathrm{2ph}}$ denotes the second two-phase interfaces
participating on the interpolation wedge $W_i$. Since the last two estimates are
exactly what is needed, we may thus conclude the proof of Lemma~\ref{lem:coercivityLenghtXi}.
\end{proof}

For a global definition of the velocity field $B$, we proceed analogously, i.e.,
we first provide a definition for local velocity fields $B^k$ for each topological
feature $\mathcal{T}_k$ and then glue them together by means of the partition of unity
$(\eta,\eta_1,\ldots,\eta_K)$ from Lemma~\ref{LemmaPartitionOfUnity}. Recall from Section~\ref{SectionLocalConstructionsTriod}
that $B^k$ necessarily has a tangential component for $k\in\mathcal{K}^{\mathrm{3j}}$. We therefore
want to exploit the freedom in choosing the tangential component for $B^k$
whenever $k\in\mathcal{K}^{\mathrm{2ph}}$, see Section~\ref{SectionLocalConstructionsTwoPhase}.

\begin{lemma}\label{LemmaLocalConstructionVelocityNetwork}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution}.
Let $(\eta,\eta_1,\ldots,\eta_K)$ be a partition of unity as constructed in
Lemma~\ref{LemmaPartitionOfUnity}. Then for every topological feature $k\in\{1,\ldots,K\}$
we may define a vector field
\begin{align*}
B^k\colon\bigcup_{t\in [0,T_{\mathrm{strong}})}\supp\eta_k(\cdot,t)\times\{t\}&\mapsto\Rd[2]
\end{align*}
based on the local velocity fields constructed in Section~\ref{SectionLocalConstructionsTwoPhase}
resp.\ Section~\ref{SectionLocalConstructionsTriod} and satisfying the following:
\begin{itemize}[leftmargin=0.7cm]
\item[i)] It holds $B^k\in L^\infty_tW^{1,\infty}_x$.
\item[ii)] Let $k\in\mathcal{K}^{\mathrm{3j}}$ and $m\in\mathcal{K}^{\mathrm{2ph}}$, such that the two-phase interface
					 $\mathcal{T}_m={\bar{I}}_{i,j}$ has an endpoint at the triple junction $\mathcal{T}_k$.
					 Then, it holds
					 \begin{align}\label{ConsistentTangentialComponent}
					 B^k &= B^m + O(\dist(\cdot,\mathcal{T}_m)),
					 \\
					 \label{ConsistentNormalTangentialDerivative}
					 \bar{\vec{n}}_{i,j}\cdot(\bar{\tau}_{i,j}\cdot\nabla)B^k
					 &=\bar{\vec{n}}_{i,j}\cdot(\bar{\tau}_{i,j}\cdot\nabla)B^m
					 + O(\dist(\cdot,\mathcal{T}_m)),
					 \\
					 \label{VanishingTangentialTangentialDerivative}
					 \bar{\tau}_{i,j}\cdot(\bar{\tau}_{i,j}\cdot\nabla)B^k
					 &= O(\dist(\cdot,\mathcal{T}_m)),
					 \\
					 \label{VanishingTangentialTangentialDerivative2}
					 \bar{\tau}_{i,j}\cdot(\bar{\tau}_{i,j}\cdot\nabla)B^m
					 &= O(\dist(\cdot,\mathcal{T}_m))
					 \end{align}
					 on $\supp\eta_k\cap\supp\eta_m$.
\end{itemize}
\end{lemma}

\begin{proof}
Let $k\in\mathcal{K}^{\mathrm{3j}}$. We define $B^k$ on the support of $\eta_k$ by means of Proposition~\ref{prop:xi_triple_junction}.
Note that by the localization property \textit{iii)} from Lemma~\ref{LemmaPartitionOfUnity}
of the partition of unity $(\eta_1,\ldots,\eta_K)$,
we are indeed in the setting of Section~\ref{SectionLocalConstructionsTriod}, so $B^k$ is well-defined.

Now, let $k\in\mathcal{K}^{\mathrm{2ph}}$. In principle, we then define $B^k$ on the support of $\eta_k$
by means of Lemma~\ref{LemmaBoundsLocalConstructionsTwoPhase}. However, in order to
ensure that \eqref{ConsistentTangentialComponent}--\eqref{VanishingTangentialTangentialDerivative}
hold true, we have to comment on the precise choice of the tangential component for $B^k$.
To this end, let $\widetilde r_c$ be the localization parameter of Lemma~\ref{LemmaPartitionOfUnity}.
Recall that $B_{2\widetilde r_c}(p_l)\cap B_{2\widetilde r_c}(p_m)=\emptyset$
by \eqref{LocalizationTwoTripleJunctions} as well as $\supp\eta_l\subset B_{\widetilde r_c}(p_l)$
by \eqref{LocalizationTripleJunction} for all triple junctions $l,m\in\mathcal{K}^{\mathrm{3j}}$ with $l\neq m$.
Let $\theta$ be a smooth cutoff function with $\theta(r)=1$ for $|r|\leq 1$ and $\theta\equiv 0$
for $|r|\geq 2$. For a triple junction $l\in\mathcal{K}^{\mathrm{3j}}$ and a two-phase interface $k\in\mathcal{K}^{\mathrm{2ph}}$
with an endpoint at $\mathcal{T}_{l}$, we denote by $\alpha_{k,l}$ the function which
is defined on $B_{2\widetilde r_c}(p_l)\cap\supp\eta_k$ and determined by the
equation \eqref{ODE_alpha}.

If the two-phase interface $\mathcal{T}_k$ has no endpoint at a triple junction, we define
$B^k$ on the support of $\eta_k$ by means of Lemma~\ref{LemmaBoundsLocalConstructionsTwoPhase}
with $\alpha:= 0$. If the two-phase interface $\mathcal{T}_k$ has exactly one of its endpoints
at a triple junction $\mathcal{T}_l$, we define $B^k$ on the support of $\eta_k$
by means of Lemma~\ref{LemmaBoundsLocalConstructionsTwoPhase} with
$\alpha:=\theta\big(\frac{\dist(\cdot,\mathcal{T}_l)}{\widetilde r_c}\big)\alpha_{k,l}$.
Finally, if the two-phase interface $\mathcal{T}_k$ has its two endpoints at
two distinct triple junctions $\mathcal{T}_l$ and $\mathcal{T}_m$, we set
$\alpha:=\theta\big(\frac{\dist(\cdot,\mathcal{T}_l)}{\widetilde r_c}\big)\alpha_{k,l}
+\theta\big(\frac{\dist(\cdot,\mathcal{T}_m)}{\widetilde r_c}\big)\alpha_{k,m}$
and define $B^k$ on the support of $\eta_k$ by means of Lemma~\ref{LemmaBoundsLocalConstructionsTwoPhase}
with this choice for the tangential component.

From these definitions, it is clear that \eqref{ConsistentTangentialComponent} is satisfied.
Moreover, it is an immediate consequence of the
equations \eqref{ODE_alpha} that \eqref{ConsistentNormalTangentialDerivative}
resp.\ \eqref{VanishingTangentialTangentialDerivative} hold true. Finally,
\eqref{VanishingTangentialTangentialDerivative2} is satisfied
as required on $\supp\eta_k\cap\supp\eta_m$, since by the localization
property \eqref{LocalizationTripleJunctionTwoPhase} we have
$\supp\eta_k\cap\supp\eta_m\subset B_{\widetilde r_c}(\mathcal{T}_k)$
entailing that the term corresponding to the gradient hitting the cutoff $\theta$
vanishes in $\supp\eta_k\cap\supp\eta_m$. In particular, the estimate \eqref{VanishingTangentialTangentialDerivative2}
then follows along the same lines as \eqref{VanishingTangentialTangentialDerivative}.
This concludes the proof of Lemma~\ref{LemmaLocalConstructionVelocityNetwork}.
\end{proof}

\begin{construction}\label{DefinitionGlobalB}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution} on some time interval $[0,T_{\mathrm{strong}})$.
Let $(\eta,\eta_1,\ldots,\eta_K)$ be a partition of unity as constructed in
Lemma~\ref{LemmaPartitionOfUnity}. Let the local vector fields $B^k$
be given as in Lemma~\ref{LemmaLocalConstructionVelocityNetwork}. We then define
\begin{align}\label{GlobalVelocityField}
B(x,t):=\sum_{k=1}^K \eta_k(x,t) B^k(x,t)
\end{align}
for all $x\in \Rd[2]$ and all $t\in [0,T_{\mathrm{strong}})$.
\end{construction}

\subsection{Compatibility of the local constructions}\label{SectionCompatibilityBoundsGluing}
Equipped with the definition of the global velocity field $B$, we may now prove the
required bound on the advective derivative of the localization functions $\eta_k$ from
Lemma~\ref{LemmaPartitionOfUnity}.

\begin{lemma}\label{LemmaAdvectionPartitionOfUnity}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution} on some time interval $[0,T_{\mathrm{strong}})$.
Let $(\eta,\eta_1,\ldots,\eta_K)$ be a partition of unity as constructed in
Lemma~\ref{LemmaPartitionOfUnity}. Let the local vector fields $B^k$
be given as in Lemma~\ref{LemmaLocalConstructionVelocityNetwork}, and let the
global velocity field $B$ be defined as in Definition~\ref{DefinitionGlobalB}.
Then, the localization functions $(\eta,\eta_1,\ldots,\eta_K)$ satisfy
\begin{align}
\label{LocalizationBound}
|\partial_t\eta_m|^2+|\nabla\eta_m|^2
&\leq C(\dist^2(x,\mathcal{T}_{m}(t))\wedge 1) + C\eta_m,
\\
\label{BulkLocalizationBound}
|\partial_t\eta|^2+|\nabla\eta|^2
&\leq C(\dist^2(x,\mathcal{T}_{m}(t))\wedge 1),
\\
\label{AdvectionEquationLocalization}
|\partial_t\eta_m+(B\cdot\nabla)\eta_m|
&\leq C(\dist^2(x,\mathcal{T}_{m}(t))\wedge 1) + C\eta_m,
\\
\label{AdvectionEquationBulkLocalization}
|\partial_t\eta+(B\cdot\nabla)\eta|
&\leq C(\dist^2(x,\mathcal{T}_{m}(t))\wedge 1),
\end{align}
for all $(x,t)\in \Rd[2]\times [0,T_{\mathrm{strong}})$, all $i,j\in\{1,\ldots,P\}$ with $i\neq j$ and
all two-phase interfaces $m\in\mathcal{K}^{\mathrm{2ph}}\subset\{1,\ldots,K\}$.
\end{lemma}

\begin{proof}
We split the proof in two parts, first establishing the asserted estimates away from triple
junctions and second in the vicinity of triple junctions.

\textit{Step 1: Estimates away from triple junctions.}
Let $m\in\mathcal{K}^{\mathrm{2ph}}$ refer to a two-phase interface $\mathcal{T}_m\subset {\bar{I}}_{p,l}$.
Because of \eqref{DefinitionAuxiliaryCutoffInterface} and \eqref{DefinitionEtaTwoPhaseAwayTripleJunction},
i.e., $\eta_m=\zeta^{\mathrm{2ph}}_{p,l}$, it holds $(\partial_t\eta_m,\nabla\eta_m)=0$ on
$\mathcal{T}_m\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}\supp\eta_k$.
Hence, by the regularity of the localization function $\eta_m$, we obtain
\begin{align*}
|\partial_t\eta_m|^2+|\nabla\eta_m|^2\leq C\dist^2(x,\mathcal{T}_{m}(t))
\quad\text{on}\quad \supp\eta_m\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}\supp\eta_k.
\end{align*}
The bound \eqref{BulkLocalizationBound} follows from this, since
$1-\eta=\eta_m$ on $\supp\eta_m\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}\supp\eta_k$.

Furthermore, on $\supp\eta_m\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}\supp\eta_k$ we have $B=\eta_m B_m$.
Exploiting \eqref{TransportSignedDistanceTwoPhase} as well as
\eqref{DefinitionAuxiliaryCutoffInterface} and $\eta_m=\zeta^{\mathrm{2ph}}_{p,l}$,
we deduce that on $\supp\eta_m\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}\supp\eta_k$
\begin{align*}
\partial_t\eta_m = -(B^m\cdot\nabla)\eta_m = -(B\cdot\nabla)\eta_m - (1-\eta_m)(B_m\cdot\nabla)\eta_m.
\end{align*}
Hence, the desired estimate
\begin{align*}
|\partial_t\eta_m+(B\cdot\nabla)\eta_m|
&\leq C\dist^2(x,\mathcal{T}_{m}(t))
\quad\text{on}\quad \supp\eta_m\setminus\bigcup_{k\in\mathcal{K}^{\mathrm{3j}}}\supp\eta_k
\end{align*}
holds true. The same is then also true for $\eta=1-\eta_m$.

\textit{Step 2: Estimates in the vicinity of triple junctions.}
Now, assume that the two-phase interface $\mathcal{T}_m$ has an endpoint at the triple
junction $\mathcal{T}_k=\{p_k\},\,k\in\mathcal{K}^{\mathrm{3j}}$. By \eqref{LocalizationTwoTripleJunctions}
and \eqref{LocalizationTripleJunctionTwoPhase}, it suffices to prove \eqref{LocalizationBound}
and \eqref{AdvectionEquationLocalization} on $B_{\widetilde r_c}(\mathcal{T}_{k})\cap (W_{p,l}\cup W_{p}\cup W_{l})$. By
\eqref{DefinitionAuxiliaryCutoffHalfSpace} resp.\ \eqref{DefinitionAuxiliaryCutoffInterface}, we know that
$(\partial_t\zeta^{\mathrm{2ph}}_{p,l},\nabla\zeta^{\mathrm{2ph}}_{p,l})=0$ on $\mathcal{T}_m\cap B_{\widetilde r_c}(\mathcal{T}_{k})$
resp.\ $(\partial_t\zeta^{\mathrm{3j}}_{p,l},\nabla\zeta^{\mathrm{3j}}_{p,l})=0$ at the triple junction $\mathcal{T}_k$.
Since on the wedge $W_{p,l}$ the distance to the triple junction is comparable to the
distance of the respective halfspaces in \eqref{DefinitionAuxiliaryCutoffHalfSpace},
we obtain together with \eqref{DefinitionEtaTwoPhaseWedgeInterface} the estimate
\begin{align*}
|\partial_t\eta_m|^2+|\nabla\eta_m|^2
&\leq C\dist^2(\cdot,\mathcal{T}_k)\zeta^{\mathrm{2ph}}_{p,l} + C\dist^2(\cdot,\mathcal{T}_m)(1-\zeta^{\mathrm{3j}}_{p,l})
\\&
\leq C(1-\zeta^{\mathrm{3j}}_{p,l})\zeta^{\mathrm{2ph}}_{p,l} + C\dist^2(\cdot,\mathcal{T}_m)
\\&
\leq C\eta_m + C\dist^2(\cdot,\mathcal{T}_m)
\quad\text{on}\quad \supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p,l}.
\end{align*}
Moreover, since $1-\eta=\eta_k+\eta_m=\zeta^{\mathrm{2ph}}_{p,l}$ on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p,l}$
we also get \eqref{BulkLocalizationBound}.
For the estimate on the advective derivative, we first note that $B=\eta_mB_m+\eta_kB_k$
on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p,l}$. Hence, by \eqref{TransportSignedDistanceTwoPhase}
we obtain as before
\begin{align*}
\partial_t\zeta^{\mathrm{2ph}}_{p,l}
&= -(B^m\cdot\nabla)\zeta^{\mathrm{2ph}}_{p,l}
\\&
= -(B\cdot\nabla)\zeta^{\mathrm{2ph}}_{p,l} - (1{-}\eta_m{-}\eta_k)(B^m\cdot\nabla)\zeta^{\mathrm{2ph}}_{p,l}
-\eta_k\big((B^m{-}B^k)\cdot\nabla\big)\zeta^{\mathrm{2ph}}_{p,l}
\end{align*}
on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p,l}$. In particular, because of
$|\nabla\zeta^{\mathrm{2ph}}_{p,l}|\leq C|\dist(\cdot,\mathcal{T}_m)|$ and \eqref{ConsistentTangentialComponent}
this entails
\begin{align*}
\big|\partial_t\zeta^{\mathrm{2ph}}_{p,l}+(B\cdot\nabla)\zeta^{\mathrm{2ph}}_{p,l}\big|
\leq C\dist^2(\cdot,\mathcal{T}_m)
\end{align*}
on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p,l}$. Since $1-\eta=\eta_k+\eta_m=\zeta^{\mathrm{2ph}}_{p,l}$
on this set, see \eqref{DefinitionEtaTripleJunctionWedgeInterface} and \eqref{DefinitionEtaTwoPhaseWedgeInterface},
this already proves \eqref{AdvectionEquationBulkLocalization} on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p,l}$.
To obtain a bound on the advective derivative of $\eta_m$ itself, we compute $\partial_t\zeta^{\mathrm{3j}}_{p,l}$.

To this end, observe first that one may express the signed distance to the halfspaces in
\eqref{DefinitionAuxiliaryCutoffHalfSpace} as follows: denoting by $\vec{h}_p(t)$
the unit normal vector of $\partial\mathbb{H}_p$ into $\mathbb{H}_p$ (which may be expressed in terms of
a fixed linear combination of the unit normals at the triple junction), we may rewrite
(recall that $\mathcal{T}_k=\{p_k\}$)
\begin{align*}
\sigdist(x,\mathbb{H}_p(t)) = -(x-p_k(t))\cdot\vec{h}_p(t)
\end{align*}
and similarly for $\sigdist(x,\mathbb{H}_l(t))$. Hence, we obtain from
$\frac{\mathrm{d}}{\mathrm{d}t}p_k(t)=B(p_k(t),t)$ that
\begin{align*}
\partial_t\zeta^{\mathrm{3j}}_{p,l} &= - (B\cdot\nabla)\zeta^{\mathrm{3j}}_{p,l}
- \big(\big(B(p_k(t),t){-}B\big)\cdot\nabla\big)\zeta^{\mathrm{3j}}_{p,l}
\\&~~~
- \zeta^{\mathrm{3j}}\Big(\frac{\sigdist(x,\mathbb{H}_{l}(t))}{c_2\widetilde r_c}\Big)
	(\zeta^{\mathrm{3j}})'\Big(\frac{\sigdist(x,\mathbb{H}_{p}(t))}{c_2\widetilde r_c}\Big)
	(x{-}p_k(t))\cdot\frac{\mathrm{d}}{\mathrm{d}t}\vec{h}_p(t)
\\&~~~
- \zeta^{\mathrm{3j}}\Big(\frac{\sigdist(x,\mathbb{H}_{p}(t))}{c_2\widetilde r_c}\Big)
	(\zeta^{\mathrm{3j}})'\Big(\frac{\sigdist(x,\mathbb{H}_{l}(t))}{c_2\widetilde r_c}\Big)
	(x{-}p_l(t))\cdot\frac{\mathrm{d}}{\mathrm{d}t}\vec{h}_l(t).
\end{align*}
Since $|B(p_k(t),t){-}B|\leq C|x{-}p_k(t)|$, $\frac{\mathrm{d}}{\mathrm{d}t}\vec{h}_p(t),
\frac{\mathrm{d}}{\mathrm{d}t}\vec{h}_l(t)\in L^\infty_t$
as well as $(\zeta^{\mathrm{3j}})'(0)=0$, we thus obtain the estimate
\begin{align}\label{AdvectionLocalizationTripleJunction}
\big|\partial_t\zeta^{\mathrm{3j}}_{p,l}+(B\cdot\nabla)\zeta^{\mathrm{3j}}_{p,l}\big|
\leq C|x-p_k|^2 \leq C(1-\zeta^{\mathrm{3j}}_{p,l})
\end{align}
on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p,l}$, since on this set
the distance to the triple junction is comparable to the
distance of the respective halfspaces in \eqref{DefinitionAuxiliaryCutoffHalfSpace}.
Hence, by taking together the bounds for the advective derivative of $\zeta^{\mathrm{2ph}}_{p,l}$ and $\zeta^{\mathrm{3j}}_{p,l}$,
we infer by using also \eqref{DefinitionEtaTwoPhaseWedgeInterface} that
\begin{align*}
|\partial_t\eta_m+(B\cdot\nabla)\eta_m| \leq C\dist^2(\cdot,\mathcal{T}_m) + C\eta_m
\end{align*}
holds true on $\supp\eta_m\cap B_{\widetilde r_c}(p_k)\cap W_{p,l}$ as required.

For the required estimates on the interpolation wedges, we make use of \eqref{DefinitionEtaTwoPhaseInterpolationWedge},
the bounds $|1-\zeta^{\mathrm{3j}}_{p,l}|\leq C|\dist(\cdot,\mathcal{T}_k)|^2$
and $|\partial_t\zeta^{\mathrm{2ph}}_{p,l}|^2+|\nabla\zeta^{\mathrm{2ph}}_{p,l}|^2\leq C|\dist(\cdot,\mathcal{T}_k)|^2$
as well as \eqref{boundslambda1} to infer first that
\begin{align*}
&|\partial_t\eta_m|^2+|\nabla\eta_m|^2
\\&
\leq C\dist^2(\cdot,\mathcal{T}_k)
\leq C\dist^2(\cdot,\mathcal{T}_m)
\quad\text{on}\quad \supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap (W_{p}\cup W_{l}).
\end{align*}
The last inequality is again due to the fact that on an interpolation wedge, the distance
to the triple junction is comparable to the distance of a two-phase interface present there.
Moreover, observe that $1-\eta=\lambda_{p}^{l}\zeta^{\mathrm{2ph}}_{p,l}+\lambda_{p}^{n}\zeta^{\mathrm{2ph}}_{p,n}$
holds true on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p}$, where we assume
that $n\in\{1,\ldots,P\}$ is the phase present at the triple junction $\mathcal{T}_k$.
Hence, the estimates $|\zeta^{\mathrm{2ph}}_{p,l}{-}\zeta^{\mathrm{2ph}}_{p,n}|\leq C|\dist(\cdot,\mathcal{T}_k)|^2$
and \eqref{boundslambda1} then entail \eqref{BulkLocalizationBound} on
$\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p}$.
The corresponding bound on the other interpolation wedges follows analogously.

For the advective derivative, we first compute
\begin{align*}
\partial_t\zeta^{\mathrm{2ph}}_{p,l}
&= -(B^m\cdot\nabla)\zeta^{\mathrm{2ph}}_{p,l}
\\&
= -(B\cdot\nabla)\zeta^{\mathrm{2ph}}_{p,l} - \big((B^m{-}B^k)\cdot\nabla\big)\zeta^{\mathrm{2ph}}_{p,l}
-\big((B^k{-}B)\cdot\nabla\big)\zeta^{\mathrm{2ph}}_{p,l}
\end{align*}
on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap (W_{p}\cup W_{l})$.
Hence, it follows from the bounds $|B{-}B^k|+|B^m{-}B^k|\leq C|\dist(\cdot,\mathcal{T}_k)|$
and $|\nabla\zeta^{\mathrm{2ph}}_{p,l}|\leq C|\dist(\cdot,\mathcal{T}_k)|$ that
\begin{align*}
\big|\partial_t\zeta^{\mathrm{2ph}}_{p,l}+(B\cdot\nabla)\zeta^{\mathrm{2ph}}_{p,l}\big|
\leq C\dist^2(\cdot,\mathcal{T}_m)
\end{align*}
on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap (W_{p}\cup W_{l})$.
From this, we again already obtain \eqref{AdvectionEquationBulkLocalization}.
Indeed, first recall that $1-\eta=\lambda_{p}^{l}\zeta^{\mathrm{2ph}}_{p,l}+\lambda_{p}^{n}\zeta^{\mathrm{2ph}}_{p,n}$
holds true on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p}$, where we assume
that $n\in\{1,\ldots,P\}$ is the third phase present at the triple junction $\mathcal{T}_k$.
Moreover, we have $|\zeta^{\mathrm{2ph}}_{p,l}{-}\zeta^{\mathrm{2ph}}_{p,n}|\leq C|\dist(\cdot,\mathcal{T}_k)|^2$
on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p}$. Hence, together with
\eqref{advectionlambda}, we obtain $|\partial_t\eta{+}(B\cdot\nabla)\eta|\leq C\dist^2(\cdot,\mathcal{T}_m)$
on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap W_{p}$. The corresponding bound on
the other interpolation wedges follows analogously.

Furthermore, it follows from the argument leading to the first inequality in
\eqref{AdvectionLocalizationTripleJunction} that
\begin{align*}
\big|\partial_t\zeta^{\mathrm{3j}}_{p,l}+(B\cdot\nabla)\zeta^{\mathrm{3j}}_{p,l}\big|
\leq C\dist^2(\cdot,\mathcal{T}_m)
\end{align*}
on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap (W_{p}\cup W_{l})$. Finally,
since $|1{-}\zeta^{\mathrm{3j}}_{p,l}|\leq C|\dist(\cdot,\mathcal{T}_k)|^2$, we may
deduce from \eqref{advectionlambda}, \eqref{DefinitionEtaTwoPhaseInterpolationWedge}
together with the bounds for the advective derivative of $\zeta^{\mathrm{2ph}}_{p,l}$ and $\zeta^{\mathrm{3j}}_{p,l}$ that
\begin{align*}
|\partial_t\eta_m+(B\cdot\nabla)\eta_m| \leq C\dist^2(\cdot,\mathcal{T}_m)
\end{align*}
holds true on $\supp\eta_m\cap B_{\widetilde r_c}(\mathcal{T}_k)\cap (W_{p}\cup W_{l})$.
This concludes the proof.
\end{proof}

We next prove several useful compatibility bounds in the gluing regions for the local constructions
associated to the different topological features of a network, i.e.,
triple junctions and smooth two-phase interfaces. These technical estimates will be needed in order to derive
the estimates \eqref{TransportEquationXi}--\eqref{Dissip} for the global constructions from
the corresponding ones for the local constructions in Lemma~\ref{LemmaBoundsLocalConstructionsTwoPhase}
and Proposition~\ref{prop:xi_triple_junction}.

\begin{lemma}\label{LemmaCompatibilityLocalConstructions}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution}.
Let $(\eta,\eta_1,\ldots,\eta_K)$ be a partition of unity as constructed in
Lemma~\ref{LemmaPartitionOfUnity}. Let $\xi^k_{i,j}$ be the local vector fields
from Lemma~\ref{LemmaLocalConstructionsNetwork} as well as $B^k$ be the local velocity fields
from Lemma~\ref{LemmaLocalConstructionVelocityNetwork}. Let $\xi_{i,j}$ be the global vector
fields from Construction~\ref{DefinitionGlobalXi}, and let $B$ be the global velocity field from
Construction~\ref{DefinitionGlobalB}. Then, the local constructions at triple junctions and
at smooth two-phase interfaces are compatible in the sense that the following estimates
are satisfied
\begin{align}
\label{CompatibilityBound1}
\big|\eta_k(\xi_{i,j}-\xi^k_{i,j})\big|
&\leq C(\dist(\cdot,{\bar{I}}_{i,j}(t))\wedge 1),
\\\label{CompatibilityBound4}
\big|\eta_k\xi_{i,j}\cdot(\xi_{i,j}-\xi^k_{i,j})\big|
&\leq C(\dist^2(\cdot,{\bar{I}}_{i,j}(t))\wedge 1),
\\\label{CompatibilityBound3}
\big|\eta_k(B^k-B)\big|
&\leq C(\dist(\cdot,{\bar{I}}_{i,j}(t))\wedge 1),
\\\label{CompatibilityBound6}
\big|\eta_k\xi_{i,j}\cdot\big((B-B^k)\cdot\nabla\big)\xi^k_{i,j}\big|
&\leq C(\dist^2(\cdot,{\bar{I}}_{i,j}(t))\wedge 1),
\\\label{CompatibilityBound5}
\big|\eta_k(\xi_{i,j}-\xi^k_{i,j})\cdot(\nabla B^k-\nabla B)^\mathsf{T}\xi^k_{i,j}\big|
&\leq C(\dist^2(\cdot,{\bar{I}}_{i,j}(t))\wedge 1)
\end{align}
for all $(x,t)\in \Rd[2]\times [0,T_{\mathrm{strong}})$, all $i,j\in\{1,\ldots,P\}$ with $i\neq j$ and
all topological features $k\in\{1,\ldots,K\}$. If we restrict to triple junctions,
we in addition have the following two slightly refined bounds
\begin{align}
\label{RefinedCompatibilityBound1}
\chi_{\supp\eta_k}|\xi^k_{i,j}{-}\xi_{i,j}|^2 +
\chi_{\supp\eta_k}|\xi_{i,j}\cdot(\xi^k_{i,j}{-}\xi_{i,j})|
&\leq C(\dist^2(\cdot,{\bar{I}}_{i,j}(t))\wedge 1),
\\\label{RefinedCompatibilityBound2}
\chi_{\supp\eta_k}|B-B^k|
&\leq C(\dist(\cdot,{\bar{I}}_{i,j}(t))\wedge 1)
\end{align}
being valid for all $(x,t)\in \Rd[2]\times [0,T_{\mathrm{strong}})$, all $i,j\in\{1,\ldots,P\}$ with $i\neq j$
and all triple junctions $k\in\mathcal{K}^{\mathrm{3j}}\subset\{1,\ldots,K\}$.
\end{lemma}

For the proof of Lemma~\ref{LemmaCompatibilityLocalConstructions}, recall that we decomposed
$\{1,\ldots,K\}=:\mathcal{K}^{\mathrm{2ph}}\cup\mathcal{K}^{\mathrm{3j}}$, with $\mathcal{K}^{\mathrm{3j}}$ enumerating the triple junctions $p_k$ present
in the strong solution, whereas $\mathcal{K}^{\mathrm{2ph}}$ enumerates the space-time connected components of the
smooth two-phase interfaces ${\bar{I}}_{i,j}$.
We also put $\mathcal{T}_k:=\{p_{k}\}$, if $k\in\mathcal{K}^{\mathrm{3j}}$, or $\mathcal{T}_k\subset {\bar{I}}_{i,j}$
for the corresponding space-time connected component $k\in\mathcal{K}^{\mathrm{2ph}}$ of a two-phase interface ${\bar{I}}_{i,j}$.

\begin{proof}[Proof of \eqref{CompatibilityBound1}, \eqref{CompatibilityBound4} and \eqref{RefinedCompatibilityBound1}]
Let $i,j\in\{1,\ldots,P\}$ be fixed with $i\neq j$. We will focus in the following
on deriving \eqref{CompatibilityBound4}; the estimates \eqref{CompatibilityBound1} and \eqref{RefinedCompatibilityBound1}
follow in the process. We start by plugging in the definition \eqref{GlobalXi} of $\xi_{i,j}$
and exploiting the fact that $\eta\leq C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)$
\begin{align*}
\eta_k\xi_{i,j}\cdot(\xi_{i,j}-\xi^k_{i,j}) &= -\eta\eta_k\xi_{i,j}\cdot\xi^k_{i,j} +
\sum_{l=1,l\neq k}^K\eta_k\eta_l\xi_{i,j}\cdot(\xi^l_{i,j}-\xi^k_{i,j})
\\&
= \sum_{m=1}^K\sum_{l=1,l\neq k}^K\eta_k\eta_l\eta_m
\xi^m_{i,j}\cdot(\xi^l_{i,j}-\xi^k_{i,j})
+ O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}

\textit{Step 1: Case $k\in\mathcal{K}^{\mathrm{3j}}$.} We consider first the case that $k\in\mathcal{K}^{\mathrm{3j}}$. Since $l\neq k$,
we may restrict ourselves to the case that $l\in\mathcal{K}^{\mathrm{2ph}}$. Indeed, by the localization
property \eqref{LocalizationTwoTripleJunctions} of the partition of unity $(\eta,\eta_1,\ldots,\eta_K)$ we have
$\supp\eta_k\cap\supp\eta_l = \emptyset$ for $k,l\in\mathcal{K}^{\mathrm{3j}}$ such that $l\neq k$.
Moreover, we may assume that $\mathcal{T}_l\subset {\bar{I}}_{i,j}$, since otherwise
either phase $i$ or $j$ is not present at $\mathcal{T}_{l}$, i.e., we can conclude directly
by means of the estimate
\begin{align}\label{auxCompatibilityWrongTwoPhase}
\eta_l \leq C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align}
Finally, we may restrict ourselves to the case that the interface $\mathcal{T}_l\subset {\bar{I}}_{i,j}$
ends at the fixed triple junction $\mathcal{T}_{k}=\{p_k\}$, since otherwise again
$\supp\eta_k\cap\supp\eta_l = \emptyset$ due to the localization
property \textit{v)} from Lemma~\ref{LemmaPartitionOfUnity}
of the partition of unity $(\eta,\eta_1,\ldots,\eta_K)$.

If $m\in\mathcal{K}^{\mathrm{3j}}$ it suffices to consider $m=k$, since otherwise $\supp\eta_m\cap\supp\eta_k=\emptyset$
by the localization property \eqref{LocalizationTwoTripleJunctions}
of the partition of unity $(\eta,\eta_1,\ldots,\eta_K)$. Moreover,
if $m\in\mathcal{K}^{\mathrm{2ph}}$ we may restrict ourselves to the case that $m=l$, i.e., in particular
$\mathcal{T}_m\subset {\bar{I}}_{i,j}$. Indeed, if this would not be the true then
either $\mathcal{T}_m$ is not a connected component of ${\bar{I}}_{i,j}$ in which
case we can argue by the analogous bound to \eqref{auxCompatibilityWrongTwoPhase}.
Or $\mathcal{T}_m\subset {\bar{I}}_{i,j}$, in which case we would necessarily
have $\supp\eta_m\cap\supp\eta_l=\emptyset$. In the following, we will now
consider first $m\in\mathcal{K}^{\mathrm{2ph}}$, i.e., $m=l$.

By assumption, the phases $i$ and $j$ are present at
the triple junction $\mathcal{T}_{k}$. Let $p\in\{1,\ldots,P\}\setminus\{i,j\}$
denote the third phase being present at $\mathcal{T}_{k}$.
Due to the localization property \eqref{LocalizationTripleJunctionTwoPhase}
of the partition of unity $(\eta,\eta_1,\ldots,\eta_K)$,
we know that $\supp\eta_k\cap\supp\eta_l$ decomposes into three wedges.
One of them (denoted in the following by $W_{i,j}$)
contains the interface $\mathcal{T}_l\subset {\bar{I}}_{i,j}$, whereas the other two are the interpolation
wedges (denoted in the following by $W_{i}$ resp.\ $W_{j}$) adjacent to the wedge $W_{i,j}$.

Recall from the proof of Proposition~\ref{prop:xi_triple_junction} that we defined
$\xi^k_{i,j}:=|\bar\xi^{\,k}_{i,j}|^{-1}\bar\xi^{\,k}_{i,j}$, where $\bar\xi^{\,k}_{i,j}$
is the local vector field defined by \eqref{def:xi_triple_junction}, and that
$|\bar\xi^{\,k}_{i,j}|\geq\frac{1}{2}$ on its domain of definition containing $\supp\eta_k$.
Now, because of
\begin{align*}
\xi^k_{i,j}-\bar\xi^{\,k}_{i,j} =
\frac{1-|\bar\xi^{\,k}_{i,j}|}{|\bar\xi^{\,k}_{i,j}|}\bar\xi^{\,k}_{i,j}
= \frac{1-|\bar\xi^{\,k}_{i,j}|^2}{(1{+}|\bar\xi^{\,k}_{i,j}|)|\bar\xi^{\,k}_{i,j}|}\bar\xi^{\,k}_{i,j}
\end{align*}
we may replace $\xi^k_{i,j}$ by $\bar\xi^{\,k}_{i,j}$ at the expense of an error
of the order $O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)$ due to the estimate \eqref{LengthXiTriod}.

On the wedge $W_{i,j}$ we may insert the corresponding local definition for $\xi^l_{i,j}$ resp.\ $\bar\xi^{\,k}_{i,j}$
from \eqref{DefinitionXiTwoPhase} resp.\ \eqref{def:xi_triple_junction}. This entails
$\xi^m_{i,j}\cdot(\xi^l_{i,j}{-}\bar\xi^{\,k}_{i,j})=O(\dist^2(\cdot,\mathcal{T}_l))$ on $W_{i,j}$.
Hence, we obtain
\begin{align}\label{auxCompatibility1}
\eta_k\eta_l\eta_m|\xi^m_{i,j}\cdot(\xi^l_{i,j}-\bar\xi^{\,k}_{i,j})|
\leq
C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align}
Note that this bound not only holds true for $m=l$, but also for $m=k$ because then
$\xi^m_{i,j}=\xi^l_{i,j}+O(\dist(\cdot,\mathcal{T}_l))$ is satisfied by \eqref{def:xi_triple_junction}.
Observe also that \eqref{CompatibilityBound1} holds true, since
$\xi^l_{i,j}{-}\bar\xi^{\,k}_{i,j}=O(\dist(\cdot,\mathcal{T}_l))$ on the wedge $W_{i,j}$.

On the interpolation wedge $W_{i}$, we again plug in the local
definition for $\bar\xi^{\,k}_{i,j}$ from \eqref{def:xi_triple_junction}, i.e.,
$\bar\xi^{\,k}_{i,j}=(1{-}\lambda_i)\widetilde\xi_{i,j}+\lambda_i R_{(p,i)}\widetilde\xi_{p,i}$, and compute
\begin{align*}
\xi^l_{i,j}-\bar\xi^{\,k}_{i,j} &= (1{-}\lambda_i)(\xi^l_{i,j}-\widetilde\xi_{i,j})
+\lambda_i(\xi^l_{i,j}-R_{(p,i)}\widetilde\xi_{p,i})
\\&
=(\xi^l_{i,j}-\widetilde\xi_{i,j})
+\lambda_i(\widetilde\xi_{i,j}-R_{(p,i)}\widetilde\xi_{p,i}).
\end{align*}
Since the argument leading to \eqref{auxCompatibility1} works
without change when $\bar\xi^{\,k}_{i,j}$ is being replaced by $\widetilde\xi_{i,j}$,
the term $\xi^m_{i,j}\cdot(\xi^l_{i,j}{-}\widetilde\xi_{i,j})$ is
bounded by $O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)$.
On the other side, by the second-order compatibility \eqref{compatibility_xi_second}
of the local vector fields $\widetilde\xi_{i,j}$ resp.\ $R_{(p,i)}\widetilde\xi_{p,i}$
at the triple junction $\mathcal{T}_k=\{p_k\}$, we have the bound
$|\widetilde\xi_{i,j}-R_{(p,i)}\widetilde\xi_{p,i}|\leq
C|x-p_k|^2 \leq C(\dist^2(x,{\bar{I}}_{i,j})\wedge 1)$.
From this, we then obtain on the interpolation wedge $W_{i}$ the desired estimate
\begin{align}\label{auxCompatibility2}
\eta_k\eta_l\eta_m|\xi^m_{i,j}\cdot(\xi^l_{i,j}-\bar\xi^{\,k}_{i,j})|^2
\leq C(\dist^2(x,{\bar{I}}_{i,j})\wedge 1).
\end{align}
The bound on the interpolation wedge $W_{j}$ follows analogously. This
concludes the proof of \eqref{CompatibilityBound1} and \eqref{CompatibilityBound4}
for $k\in\mathcal{K}^{\mathrm{3j}}$. Before we proceed with the argument for $k\in\mathcal{K}^{\mathrm{2ph}}$,
let us make the following observation. In the whole preceding argument for $k\in\mathcal{K}^{\mathrm{3j}}$,
we not fully exploited the fact that we have the localization function $\eta_k$
as a prefactor. We actually only used that we have to provide a bound on $\supp\eta_k$.
Therefore, what the preceding argument actually proves is the slightly refined estimate
\eqref{RefinedCompatibilityBound1}.

\textit{Step 2: $k\in\mathcal{K}^{\mathrm{2ph}}$.} As argued before, it is then
sufficient to consider the case that $\mathcal{T}_{k}\subset {\bar{I}}_{i,j}$
since otherwise we may conclude by the corresponding bound of \eqref{auxCompatibilityWrongTwoPhase}.
If now $l\in\mathcal{K}^{\mathrm{3j}}$, it moreover suffices to restrict to the case that
$\mathcal{T}_k$ ends at the fixed triple junction $\mathcal{T}_l=\{p_l\}$.
In other words, the roles of $k$ and $l$ are reversed and we may apply the
argument from the previous case (except for \eqref{RefinedCompatibilityBound1},
but this will not be needed in the sequel anyway). Finally, if $l\in\mathcal{K}^{\mathrm{2ph}}$ and $l\neq k$,
then either $\mathcal{T}_l\subset {\bar{I}}_{i,j}$ which in turn necessarily entails $\supp\eta_k\cap\supp\eta_l=\emptyset$,
or otherwise either phase $i$ or $j$ is absent at $\mathcal{T}_l$ meaning we may rely again on the bound
\eqref{auxCompatibilityWrongTwoPhase}. This concludes the proof of \eqref{CompatibilityBound1}
and \eqref{CompatibilityBound4}.
\end{proof}

\begin{proof}[Proof of \eqref{CompatibilityBound3} and \eqref{RefinedCompatibilityBound2}]
Let $i,j\in\{1,\ldots,P\}$ be fixed with $i\neq j$. From the definition
\eqref{GlobalVelocityField} as well as $\eta\leq C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)$ it follows
\begin{align*}
\eta_k(B^k{-}B)
&=\eta\eta_kB^k +
\sum_{l=1,l\neq k}^K\eta_k\eta_l(B^k{-}B^l)
\\&
=\sum_{l=1,l\neq k}^K\eta_k\eta_l(B^k{-}B^l)
+ O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
Let now $k\in\mathcal{K}^{\mathrm{3j}}$. As argued in the proof of \eqref{CompatibilityBound1}, we may restrict
ourselves to the case that $l\in\mathcal{K}^{\mathrm{2ph}}$ such that $\mathcal{T}_{l}\subset {\bar{I}}_{i,j}$,
and the interface $\mathcal{T}_l$ ends at the fixed triple junction $\mathcal{T}_k=\{p_k\}$.
Again, $\supp\eta_k\cap\supp\eta_l$ then decomposes into three wedges $W_{i,j}$ and $W_{i}$ resp.\ $W_{j}$,
see the proof of \eqref{CompatibilityBound1}. To fix notation, let us suppose that the phase $p\in\{1,\ldots,P\}$
is, next to the phases $i$ and $j$, present at $\mathcal{T}_{k}$.

By the precise choice of the tangential component for the velocity field $B^l$
in $\supp\eta_k\cap\supp\eta_l$, we have \eqref{ConsistentTangentialComponent}.
Hence, it follows
\begin{align}\label{auxCompatibility3}
\eta_k\eta_l|(B^k-B^l)|
\leq
C(\dist(\cdot,{\bar{I}}_{i,j})\wedge 1) \quad\text{on } W_{i,j}.
\end{align}
To obtain the corresponding bound on the interpolation wedge $W_{i}$, we plug in
the local definition of $B^k$ from \eqref{def:velocity_triple_junction}, i.e.,
$B^k=(1{-}\lambda_i)\widetilde B_{(i,j)}+\lambda_{i}\widetilde B_{(i,p)}$, and compute
\begin{align}\label{auxDecompositionVelocity}
B^k-B^l &=(1{-}\lambda_i)(\widetilde B_{(i,j)}{-}B^l)+\lambda_{i}(\widetilde B_{(i,p)}{-}B^l)
\\&\nonumber
= (\widetilde B_{(i,j)}{-}B^l) + \lambda_{i}(\widetilde B_{(i,p)}{-}\widetilde B_{(i,j)}).
\end{align}
Now, since the derivation of the estimate \eqref{auxCompatibility3} works without change
when replacing the velocity field $B^k$ by $\widetilde B_{(i,j)}$, we obtain for the first
term in the above decomposition a bound of required order.
For the second term, we employ the first-order compatibility \eqref{compatibility_B_first} of the
velocity fields $\widetilde B_{(i,j)}$ resp.\ $\widetilde B_{(i,p)}$ at the triple junction
$\mathcal{T}_k=\{p_k\}$ to obtain again an estimate of required order.
All in all, we may infer that on the interpolation wedge $W_{i}$ it holds
\begin{align}\label{auxCompatibility4}
\eta_k\eta_l|(B^k-B^l)|^2 \leq C(\dist(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align}
The corresponding bound on the interpolation wedge $W_{j}$ follows along
the same lines. Hence, we proved \eqref{CompatibilityBound3} for $k\in\mathcal{K}^{\mathrm{3j}}$.
As in the proof of \eqref{CompatibilityBound1} resp.\ \eqref{CompatibilityBound4},
we make the observation that what we actually proved is the estimate \eqref{RefinedCompatibilityBound2}.
Finally, the other case $k\in\mathcal{K}^{\mathrm{2ph}}$ follows by an analogous case distinction
as at the end of the proof of \eqref{CompatibilityBound1} resp.\ \eqref{CompatibilityBound4}.
\end{proof}

\begin{proof}[Proof of \eqref{CompatibilityBound6}]
Let $i,j\in\{1,\ldots,P\}$ be fixed with $i\neq j$. Plugging in \eqref{GlobalXi}
and \eqref{GlobalVelocityField} as well as making use of
$\eta\leq C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)$ we may compute
\begin{align*}
&\eta_k\xi_{i,j}\cdot\big((B{-}B^k)\cdot\nabla\big)\xi^k_{i,j}
\\&
=-\eta\eta_k\xi_{i,j}\cdot(B^k\cdot\nabla)\xi^k_{i,j}
+\sum_{l=1,l\neq k}^K\eta_k\eta_l\xi_{i,j}\cdot\big((B^l{-}B^k)\cdot\nabla\big)\xi^k_{i,j}
\\&
=\sum_{m=1}^K\sum_{l=1,l\neq k}^K\eta_k\eta_l\eta_m\xi^m_{i,j}\cdot\big((B^l{-}B^k)\cdot\nabla\big)\xi^k_{i,j}
+O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
Let $k\in\mathcal{K}^{\mathrm{3j}}$. As it is by now routine, we may restrict
ourselves to the case that $l\in\mathcal{K}^{\mathrm{2ph}}$ such that $\mathcal{T}_{l}\subset {\bar{I}}_{i,j}$,
and the interface $\mathcal{T}_l$ ends at the fixed triple junction $\mathcal{T}_k=\{p_k\}$.
Recall that $\supp\eta_k\cap\supp\eta_l$ then decomposes into three wedges $W_{i,j}$ and $W_{i}$ resp.\ $W_{j}$,
see the proof of \eqref{CompatibilityBound1}. To fix notation, let us suppose that the phase $p\in\{1,\ldots,P\}$
is, next to the phases $i$ and $j$, present at $\mathcal{T}_{k}$.

If $m\in\mathcal{K}^{\mathrm{3j}}$ it suffices to consider $m=k$, since otherwise $\supp\eta_m\cap\supp\eta_k=\emptyset$
by the localization property \eqref{LocalizationTwoTripleJunctions}
of the partition of unity $(\eta,\eta_1,\ldots,\eta_K)$. Moreover,
if $m\in\mathcal{K}^{\mathrm{2ph}}$ we may restrict ourselves to the case that $m=l$
as argued in the proof of \eqref{CompatibilityBound1} resp.\ \eqref{CompatibilityBound4}, i.e.,
in particular $\mathcal{T}_m\subset {\bar{I}}_{i,j}$.
In the following, we will consider first $m\in\mathcal{K}^{\mathrm{2ph}}$, i.e., $m=l$.

On the wedge $W_{i,j}$, we have $\xi^l_{i,j}=\bar{\vec{n}}_{i,j}$.
By the precise choice of the tangential component for the velocity field $B^l$
in $\supp\eta_k\cap\supp\eta_l$, we have \eqref{ConsistentTangentialComponent}.
In particular, on the wedge $W_{i,j}$ it holds
$B^l-B^k=O(\dist(\cdot,\mathcal{T}_{l}))\bar{\tau}_{i,j}+O(\dist^2(\cdot,\mathcal{T}_{l}))$
due to the definition of $B^k$ in \eqref{def:velocity_triple_junction}.
Moreover, we have $\nabla\xi^k_{i,j}=O(1)\bar{\tau}_{i,j}\otimes \bar{\tau}_{i,j}
+O(1)\bar{\tau}_{i,j}\otimes\bar{\vec{n}}_{i,j} +
O(1)\bar\xi^{\,k}_{i,j}\otimes(\nabla\bar\xi^{\,k}_{i,j})^\mathsf{T}\bar\xi^{\,k}_{i,j} + O(\dist(\cdot,\mathcal{T}_{l}))$ on
the wedge $W_{i,j}$. However, since $(\nabla\bar\xi^{\,k}_{i,j})^\mathsf{T}\bar\xi^{\,k}_{i,j}=O(\dist(\cdot,\mathcal{T}_{l}))$
we get $\nabla\xi^k_{i,j}=O(1)\bar{\tau}_{i,j}\otimes \bar{\tau}_{i,j}
+O(1)\bar{\tau}_{i,j}\otimes\bar{\vec{n}}_{i,j} + O(\dist(\cdot,\mathcal{T}_{l}))$
on the wedge $W_{i,j}$. Hence, taking all of these bounds together we obtain
\begin{align}
\label{auxCompatibility5}
\eta_k\eta_l\eta_m\big|\xi^m_{i,j}\cdot\big((B^l{-}B^k)\cdot\nabla\big)\xi^k_{i,j}\big|
\leq C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align}
Note that the bound \eqref{auxCompatibility5} also holds for $m\in\mathcal{K}^{\mathrm{3j}}$ (i.e., $m=k$ as argued above),
since on the wedge $W_{i,j}$ we have $\xi^k_{i,j}=\xi^l_{i,j} + O(\dist(\cdot,\mathcal{T}_{l}))$.

On the interpolation wedge $W_{i}$, we still have $\xi^l_{i,j}=\bar{\vec{n}}_{i,j}$.
For $B^k-B^l$, we first refer to the decomposition \eqref{auxDecompositionVelocity}.
Note that apart from the first-order compatibility \eqref{compatibility_B_first} of the velocity fields $\widetilde B_{(i,j)}$
resp.\ $\widetilde B_{(i,p)}$ at the triple junction $\mathcal{T}_k=\{p_k\}$, we also know
that $\widetilde B_{(i,j)}$ resp.\ $\widetilde B_{(i,p)}$ have bounded and continuous second derivatives on $W_{i}$.
Hence, we may use the first-order compatibility of the velocity fields at the triple junction to estimate
$B^k{-}B^l=(\widetilde B_{(i,j)}{-}B^l) + O(|x{-}p_k|^2)$ on the interpolation wedge $W_{i}$.
It moreover follows from the definition of the velocity $\widetilde B_{(i,j)}$
as well as the argument for the wedge $W_{i,j}$ that
$\widetilde B_{(i,j)}{-}B^l=O(\dist(\cdot,\mathcal{T}_{l}))\bar{\tau}_{i,j}+O(\dist^2(\cdot,\mathcal{T}_{l}))$.

Next, we plug in the local definition for the vector field
$\bar\xi^{\,k}_{i,j}$ from \eqref{def:xi_triple_junction}, i.e.,
$\bar\xi^{\,k}_{i,j}=(1{-}\lambda_i)\widetilde\xi_{i,j}+\lambda_i R_{(p,i)}\widetilde\xi_{p,i}
=\widetilde\xi_{i,j}+\lambda_i(R_{(p,i)}\widetilde\xi_{p,i}{-}\widetilde\xi_{i,j})$, and compute
\begin{align*}
\nabla\bar\xi^{\,k}_{i,j}=(R_{(p,i)}\widetilde\xi_{p,i}{-}\widetilde\xi_{i,j})\otimes\nabla\lambda_i
+\nabla\widetilde\xi_{i,j}+\lambda_i\nabla(R_{(p,i)}\widetilde\xi_{p,i}{-}\widetilde\xi_{i,j}).
\end{align*}
Moreover, note that we have $\nabla|\bar\xi^{\,k}_{i,j}|^{-1}=O(1)(\nabla\bar\xi^{\,k}_{i,j})^\mathsf{T}\bar\xi^{\,k}_{i,j}$.
It then follows from the second-order compatibility \eqref{compatibility_xi_second} of the vector fields $\widetilde\xi_{i,j}$
resp.\ $R_{(p,i)}\widetilde\xi_{p,i}$ at the triple junction $\mathcal{T}_k=\{p_k\}$,
as well as the definition of $\widetilde\xi_{i,j}$ together with the argument for the
wedge $W_{i,j}$, that $\nabla\xi^k_{i,j}=O(1)\bar{\tau}_{i,j}\otimes \bar{\tau}_{i,j}
+O(1)\bar{\tau}_{i,j}\otimes\bar{\vec{n}}_{i,j}) + O(\dist(\cdot,\mathcal{T}_{l}))$
also holds on the interpolation wedge $W_{i}$. Taking all of these bounds together we may infer
\begin{align}\label{auxCompatibility6}
\eta_k\eta_l\eta_m\big|\xi^m_{i,j}\cdot\big((B^l{-}B^k)\cdot\nabla\big)\xi^k_{i,j}\big|
\leq C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)
\end{align}
on the interpolation wedge $W_{i}$. Note that the bound \eqref{auxCompatibility6} also holds
for $m\in\mathcal{K}^{\mathrm{3j}}$ (i.e., $m=k$ as argued above), since on the interpolation wedge $W_{i}$ we again have
$\xi^k_{i,j}=\xi^l_{i,j} + O(\dist(\cdot,\mathcal{T}_{l}))$. As the argument for the interpolation wedge
$W_{j}$ proceeds along the same lines, we thus have proved \eqref{CompatibilityBound6}
for $k\in\mathcal{K}^{\mathrm{3j}}$. The argument for $k\in\mathcal{K}^{\mathrm{2ph}}$ proceeds by an analogous case distinction to the one
at the end of the proof of \eqref{CompatibilityBound1}.
\end{proof}

\begin{proof}[Proof of \eqref{CompatibilityBound5}]
Let again $i,j\in\{1,\ldots,P\}$ be fixed with $i\neq j$. This time, we compute
using \eqref{GlobalXi}, \eqref{GlobalVelocityField}, $\eta\leq C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)$
as well as \eqref{LocalizationBound} and \eqref{CompatibilityBound1}
\begin{align*}
&\eta_k(\xi_{i,j}-\xi^k_{i,j})\cdot(\nabla B^k-\nabla B)^\mathsf{T}\xi^k_{i,j}
\\&
=\eta\eta_k(\xi_{i,j}{-}\xi^k_{i,j})\cdot(\nabla B^k)^\mathsf{T}\xi^k_{i,j}
+\sum_{l=1,l\neq k}^K\eta_k\eta_l(\xi_{i,j}{-}\xi^k_{i,j})\cdot
(\nabla B^k{-}\nabla B^l)^\mathsf{T}\xi^k_{i,j}
\\&~~~
+ \eta_k(\xi_{i,j}{-}\xi^k_{i,j})\cdot(\nabla\eta\otimes B)\xi^k_{i,j}
-\sum_{l=1}^K\eta_k(\xi_{i,j}{-}\xi^k_{i,j})\cdot(\nabla\eta_l\otimes (B^l{-}B))\xi^k_{i,j}
\\&
=\sum_{m=1,m\neq k}^K\sum_{l=1,l\neq k}^K\eta_k\eta_l\eta_m(\xi^m_{i,j}{-}\xi^k_{i,j})\cdot
(\nabla B^k{-}\nabla B^l)^\mathsf{T}\xi^k_{i,j}
\\&~~~
-\sum_{l=1}^K\eta_k(\xi_{i,j}{-}\xi^k_{i,j})\cdot(\nabla\eta_l\otimes (B^l{-}B))\xi^k_{i,j}
+O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}

We first argue how to estimate the last term in the previous estimate. If $l\in\mathcal{K}^{\mathrm{3j}}$,
we obtain a bound of required order by an application of H\"older's inequality
as well as the estimates \eqref{CompatibilityBound1} and \eqref{RefinedCompatibilityBound2}.
On the other side, if $l\in\mathcal{K}^{\mathrm{2ph}}$ we may argue by means of H\"older's inequality
as well as the estimates \eqref{CompatibilityBound1}, \eqref{CompatibilityBound3}
and \eqref{LocalizationBound}. Hence, we may move on with the bound
for the penultimate term.

Let $k\in\mathcal{K}^{\mathrm{3j}}$; the case $k\in\mathcal{K}^{\mathrm{2ph}}$ follows by symmetry.
We assume , without loss of generality, that $l\in\mathcal{K}^{\mathrm{2ph}}$ such that $\mathcal{T}_{l}\subset {\bar{I}}_{i,j}$
and $\mathcal{T}_{l}$ has an endpoint at the triple junction $\mathcal{T}_{k}=\{p_k\}$. We again decompose
$\supp\eta_k\cap\supp\eta_l$ into three wedges $W_{i,j}$ and $W_{i}$ resp.\ $W_{j}$,
see the proof of \eqref{CompatibilityBound1}. To fix notation, let us suppose that the phase $p\in\{1,\ldots,P\}$
is, next to the phases $i$ and $j$, present at $\mathcal{T}_{k}$.
Let $m\in\mathcal{K}^{\mathrm{2ph}}$ (for $m\in\mathcal{K}^{\mathrm{3j}}$ with $m\neq k$ we have $\supp\eta_k\cap\supp\eta_m=\emptyset$
due to the localization property \eqref{LocalizationTwoTripleJunctions}
of the partition of unity $(\eta,\eta_1,\ldots,\eta_K)$).
We may then also assume that $m=l$, as argued in the proof of \eqref{CompatibilityBound1}
resp.\ \eqref{CompatibilityBound4}.
Finally, we can replace $\xi^k_{i,j}$ by $\bar\xi^{\,k}_{i,j}$ at the expense of an error
of the order $O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)$ as in the proof of \eqref{CompatibilityBound1}
resp.\ \eqref{CompatibilityBound4}.

By Property \textit{iii)} of Lemma~\ref{LemmaLocalConstructionsNetwork},
the Lipschitz continuity of the vector fields $\bar\xi^{\,k}_{i,j}$
resp.\ $\xi^l_{i,j}$ and the definition of $\xi_{i,j}^k$ from \eqref{def:xi_triple_junction}, we have
$\xi^l_{i,j}{-}\bar\xi^{\,k}_{i,j}=O(\dist(\cdot,\mathcal{T}_{l}))\bar{\tau}_{i,j}+O(\dist^2(\cdot,\mathcal{T}_{l}))$
on the wedge $W_{i,j}$. In particular,
$\bar\xi^{\,k}_{i,j}=\bar{\vec{n}}_{i,j}+O(\dist(\cdot,\mathcal{T}_{l}))\bar{\tau}_{i,j}+O(\dist^2(\cdot,\mathcal{T}_{l}))$.
On the other side, by the precise choice of the tangential component for the velocity field $B^l$
in $\supp\eta_k\cap\supp\eta_l$, we have \eqref{ConsistentNormalTangentialDerivative} and
\eqref{VanishingTangentialTangentialDerivative}. Hence, by the local definition of
the velocity field $B^k$ from \eqref{def:velocity_triple_junction} it holds
\begin{align*}
(\nabla B^k{-}\nabla B^l)^\mathsf{T}
&=O(1)\bar{\vec{n}}_{i,j}\otimes\bar{\tau}_{i,j}
+O(\dist(\cdot,\mathcal{T}_{l}))\bar{\vec{n}}_{i,j}\otimes\bar{\vec{n}}_{i,j}
\\&~~~
+O(\dist(\cdot,\mathcal{T}_{l}))\bar\tau_{i,j}\otimes\bar{\vec{n}}_{i,j}
+O(\dist(\cdot,\mathcal{T}_{l}))\bar\tau_{i,j}\otimes\bar\tau_{i,j}
\end{align*}
on the wedge $W_{i,j}$. Note that indeed no zero-order contribution
for the $\bar{\vec{n}}_{i,j}\otimes \bar{\vec{n}}_{i,j}$ component appears since
neither $\nabla B^k$ nor $\nabla B^l$ contain one already on an individual basis.
Hence, when contracting with $\bar\xi^{\,k}_{i,j}$ and
$\xi^l_{i,j}{-}\bar\xi^{\,k}_{i,j}$ we obtain
\begin{align}\label{auxCompatibility7}
\eta_k\eta_l\eta_m\big|(\xi^m_{i,j}{-}\bar\xi^{\,k}_{i,j})\cdot
(\nabla B^k{-}\nabla B^l)^\mathsf{T}\bar\xi^{\,k}_{i,j}\big|
\leq C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)
\end{align}
on the wedge $W_{i,j}$.

Thus, we may turn to the bound on the interpolation wedge $W_{i}$.
By substitution of the local definition of $\bar\xi^{\,k}_{i,j}$ from \eqref{def:xi_triple_junction}, we first
observe that $\bar\xi^{\,k}_{i,j}=\widetilde\xi_{i,j}+\lambda_i(R_{(p,i)}\widetilde\xi_{p,i}{-}\widetilde\xi_{i,j})$.
Hence, it follows from the second-order compatibility of the
vector fields $\widetilde\xi_{i,j}$ resp.\ $R_{(p,i)}\widetilde\xi_{p,i}$ at the triple
junction $\mathcal{T}_{k}=\{p_k\}$, as well as the definition of $\widetilde\xi_{i,j}$
that $\bar\xi^{\,k}_{i,j}=\bar{\vec{n}}_{i,j}+O(\dist(\cdot,\mathcal{T}_{l}))\bar{\tau}_{i,j}+O(\dist^2(\cdot,\mathcal{T}_{l}))$.
In particular, by definition of $\xi^l_{i,j}$ from \eqref{DefinitionXiTwoPhase} it holds
$\xi^l_{i,j}{-}\bar\xi^{\,k}_{i,j}=O(\dist(\cdot,\mathcal{T}_{l}))\bar{\tau}_{i,j}+O(\dist^2(\cdot,\mathcal{T}_{l}))$.
Next, we decompose
\begin{align*}
\nabla B^k{-}\nabla B^l
&=(\widetilde B_{(i,j)}{-}\widetilde B_{(i,p)})\otimes\nabla\lambda
+(\nabla\widetilde B_{(i,j)}{-}\nabla B^l)+(1{-}\lambda)(\nabla\widetilde B_{(i,j)}{-}\nabla\widetilde B_{(i,p)}).
\end{align*}
Using the first-order compatibility of the velocity fields at the triple junction and the uniform bounds on their second spatial derivatives, we estimate
$(\nabla B^k{-}\nabla B^l)^\mathsf{T}=(\nabla\widetilde B_{(i,j)}{-}\nabla B^l)^\mathsf{T}+O(|x{-}p_k|)$.
By definition of $\widetilde B_{(i,j)}$ we also obtain as before
\begin{align*}
(\nabla\widetilde B_{(i,j)}{-}\nabla B^l)^\mathsf{T}
&=O(1)\bar{\vec{n}}_{i,j}\otimes\bar{\tau}_{i,j}
+O(\dist(\cdot,\mathcal{T}_{l}))\bar{\vec{n}}_{i,j}\otimes\bar{\vec{n}}_{i,j}
\\&~~~
+O(\dist(\cdot,\mathcal{T}_{l}))\bar\tau_{i,j}\otimes\bar{\vec{n}}_{i,j}
+O(\dist(\cdot,\mathcal{T}_{l}))\bar\tau_{i,j}\otimes\bar\tau_{i,j}.
\end{align*}
In summary, we may infer that on the interpolation wedge $W_{i}$ it holds
\begin{align}
\label{auxCompatibility8}
\eta_k\eta_l\eta_m\big|(\xi^m_{i,j}{-}\bar\xi^{\,k}_{i,j})\cdot
(\nabla B^k{-}\nabla B^l)^\mathsf{T}\bar\xi^{\,k}_{i,j}\big|
\leq C(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align}
This establishes \eqref{CompatibilityBound5} for $k\in\mathcal{K}^{\mathrm{3j}}$,
and therefore concludes the proof.
\end{proof}

\subsection{Approximate transport and mean curvature flow equations: Proof of Proposition \ref{PropositionExistenceCalibration}}\label{SectionBoundsTimeEvolutionGlobalConstruction}
We first derive the global (or network) version of the bounds from Lemma~\ref{LemmaBoundsLocalConstructionsTwoPhase}
resp.\ Proposition~\ref{prop:xi_triple_junction}, which represent the model
problem of a smooth manifold resp.\ of a regular triod evolving by mean curvature.

\begin{lemma}\label{LemmaGlobalCoercivityBounds}
Let $\bar\chi=(\bar\chi_1,\ldots,\bar\chi_P)$ be a strong solution for multiphase
mean curvature flow in the sense of Definition~\ref{DefinitionStrongSolution}.
Let $\xi_{i,j}$ be the global vector
fields from Construction~\ref{DefinitionGlobalXi}, and let $B$ be the global velocity field from
Construction~\ref{DefinitionGlobalB}.
Then we have the estimates
\begin{align}
\label{TransportEquationXiGlobal}
|\partial_t\xi_{i,j} + ( B\cdot\nabla)\xi_{i,j} + (\nabla B)^\mathsf{T}\xi_{i,j}| &\leq
C(\dist(x,{\bar{I}}_{i,j}(t))\wedge 1),
\\
\label{DissipGlobal}
|(\nabla\cdot\xi_{i,j}) +  B\cdot\xi_{i,j}| &\leq
C(\dist(x,{\bar{I}}_{i,j}(t))\wedge 1),
\\
\label{LengthConservationGlobal}
\big|\xi_{i,j}\cdot\partial_t\xi_{i,j} + \xi_{i,j}\cdot(B\cdot\nabla)\xi_{i,j}\big| &\leq
C(\dist^2(x,{\bar{I}}_{i,j}(t))\wedge 1)
\end{align}
for all $(x,t)\in \Rd[2]\times [0,T_{\mathrm{strong}})$ and all $i,j\in\{1,\ldots,P\}$ with $i\neq j$.
\end{lemma}

For the proof, recall that we decomposed
$\{1,\ldots,K\}=:\mathcal{K}^{\mathrm{2ph}}\cup\mathcal{K}^{\mathrm{3j}}$, with $\mathcal{K}^{\mathrm{3j}}$ enumerating the triple junctions $p_k$ present
in the strong solution, whereas $\mathcal{K}^{\mathrm{2ph}}$ enumerates the space-time connected components of the
smooth two-phase interfaces ${\bar{I}}_{i,j}$.
We then defined $\mathcal{T}_k:=\{p_{k}\}$, if $k\in\mathcal{K}^{\mathrm{3j}}$, or $\mathcal{T}_k\subset {\bar{I}}_{i,j}$
for the corresponding space-time connected component $k\in\mathcal{K}^{\mathrm{2ph}}$ of a two-phase interface ${\bar{I}}_{i,j}$.

\begin{proof}[Proof of \eqref{TransportEquationXiGlobal}]
Let $i,j\in\{1,\ldots,P\}$ with $i\neq j$ be fixed. If either $k\in\mathcal{K}^{\mathrm{3j}}$ such that
the interface ${\bar{I}}_{i,j}$ has an endpoint at the triple junction
$\mathcal{T}_{k}$, or $k\in\mathcal{K}^{\mathrm{2ph}}$ such that $\mathcal{T}_{k}\subset {\bar{I}}_{i,j}$,
then it follows from the local evolution equations
\eqref{TransportEquationXiTwoPhase} resp.\ \eqref{TransportEquationXiTriod}
\begin{align*}
\eta_k\partial_t\xi^k_{i,j}=-\eta_k(B^k\cdot\nabla)\xi^k_{i,j}-\eta_k(\nabla B^k)^\mathsf{T}\xi^k_{i,j}
+ O(\dist(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
On the other side, if neither of the above two mentioned cases holds true, we may
simply make use of estimate
\begin{align}\label{GlobalEquationsBoundMinorityPhases}
\eta_k\leq C(\dist(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align}
Hence, we may compute in total
\begin{align*}
\partial_t\xi_{i,j} &=
\sum_{k=1}^K\eta_k\partial_t\xi^k_{i,j} +
\sum_{k=1}^K\xi_{i,j}^k\partial_t\eta_k
\\&
=- \sum_{k=1}^K\eta_k(B^k\cdot\nabla)\xi^k_{i,j}
-\sum_{k=1}^K\eta_k(\nabla B^k)^\mathsf{T}\xi^k_{i,j}
+\sum_{k=1}^K\xi_{i,j}^k\partial_t\eta_k +
O(\dist(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
By inserting $\xi_{i,j}$ in the second term and $B$ in the first term,
we obtain from the compatibility bounds \eqref{CompatibilityBound1} and \eqref{CompatibilityBound3}
\begin{align*}
\partial_t\xi_{i,j}
&= -\sum_{k=1}^K\eta_k(B\cdot\nabla)\xi^k_{i,j}
-\sum_{k=1}^K\eta_k(\nabla B^k)^\mathsf{T}\xi_{i,j}
+\sum_{k=1}^K\xi^k_{i,j}\partial_t\eta_k
+ O(\dist(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
Employing the product rule, we further compute
\begin{align*}
&-\sum_{k=1}^K\eta_k(B\cdot\nabla)\xi^k_{i,j}
-\sum_{k=1}^K\eta_k(\nabla B^k)^\mathsf{T}\xi_{i,j}
+\sum_{k=1}^K\xi^k_{i,j}\partial_t\eta_k
\\&
=-(B\cdot\nabla)\xi_{i,j} -(\nabla B)^\mathsf{T}\xi_{i,j}
\\&~~~
+\sum_{k=1}^K(\xi^k_{i,j}{-}\xi_{i,j})\big(\partial_t\eta_k{+}(B\cdot\nabla)\eta_k\big)
+\sum_{k=1}^K\big((B^k{-}B)\cdot\xi_{i,j}\big)\nabla\eta_k
\\&~~~
-\xi_{i,j}\big(\partial_t\eta{+}(B\cdot\nabla)\eta\big)
-(B\cdot\xi_{i,j})\nabla\eta +  O(\dist(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
The terms in the last line are bounded due to \eqref{BulkLocalizationBound}
and \eqref{AdvectionEquationBulkLocalization}. For the sums in the penultimate line,
we split them by means of $\{1,\ldots,K\}=\mathcal{K}^{\mathrm{3j}}\cup\mathcal{K}^{\mathrm{2ph}}$. The sum over the two-phase
interfaces $k\in\mathcal{K}^{\mathrm{2ph}}$ may be absorbed into the error term $O(\dist(\cdot,{\bar{I}}_{i,j})\wedge 1)$ by an application
of \eqref{LocalizationBound}, \eqref{AdvectionEquationLocalization}, \eqref{CompatibilityBound1}
as well as \eqref{CompatibilityBound3}. On the other side, the sum over the triple junctions
$k\in\mathcal{K}^{\mathrm{3j}}$ is an error term of required order, since we may apply the slightly
refined compatibility bounds \eqref{RefinedCompatibilityBound1} and \eqref{RefinedCompatibilityBound2}.
This then concludes the proof of \eqref{TransportEquationXiGlobal}.
\end{proof}

\begin{proof}[Proof of \eqref{DissipGlobal}]
Let $i,j\in\{1,\ldots,P\}$ with $i\neq j$ be fixed. Following the argument
at the beginning of the proof of \eqref{TransportEquationXiGlobal}, i.e., by either
making use of the local equations \eqref{DissipTwoPhase} resp.\ \eqref{DissipTriod}
or the bound \eqref{GlobalEquationsBoundMinorityPhases}, we may compute
\begin{align*}
\nabla\cdot\xi_{i,j} &= \sum_{k=1}^K\eta_k(\nabla\cdot\xi^k_{i,j})
+ \sum_{k=1}^K(\xi^k_{i,j}\cdot\nabla)\eta_k
\\&
= -\sum_{k=1}^K\eta_k B^k\cdot\xi^k_{i,j}
+\sum_{k=1}^K(\xi^k_{i,j}\cdot\nabla)\eta_k
+  O(\dist(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
Next, we rewrite
\begin{align*}
&-\sum_{k=1}^K\eta_kB^k\cdot\xi^k_{i,j} + \sum_{k=1}^K(\xi^k_{i,j}\cdot\nabla)\eta_k
\\&
=-B\cdot\xi_{i,j}
-\sum_{k=1}^K\eta_k(B^k{-}B)\cdot\xi^k_{i,j}
+\sum_{k=1}^K\big((\xi^k_{i,j}{-}\xi_{i,j})\cdot\nabla\big)\eta_k
-(\xi_{i,j}\cdot\nabla)\eta.
\end{align*}
The second term in the latter identity is bounded by means of \eqref{CompatibilityBound3},
whereas for the last term one relies on \eqref{BulkLocalizationBound}. The term with
the sum over the topological features may be treated by splitting $\{1,\ldots,K\}=\mathcal{K}^{\mathrm{3j}}\cup\mathcal{K}^{\mathrm{2ph}}$
as at the end of the proof of \eqref{TransportEquationXiGlobal}.
Hence, we indeed obtain \eqref{DissipGlobal}.
\end{proof}

\begin{proof}[Proof of \eqref{LengthConservationGlobal}]
Let again $i,j\in\{1,\ldots,P\}$ with $i\neq j$ be fixed. Adapting the argument
at the beginning of the proof of \eqref{TransportEquationXiGlobal}, i.e., either
making use of the local equations \eqref{LengthConservationTwoPhase}
resp.\ \eqref{LengthConservationXiTriodNormalized} or exploiting the bound \eqref{GlobalEquationsBoundMinorityPhases},
we have
\begin{align*}
\xi_{i,j}\cdot\partial_t\xi_{i,j} &= \sum_{k=1}^K\eta_k\xi_{i,j}\cdot\partial_t\xi^k_{i,j}
+\sum_{k=1}^K\xi_{i,j}\cdot\xi^k_{i,j}\partial_t\eta_k
\\&
=-\sum_{k=1}^K\eta_k\xi^k_{i,j}\cdot(B^k\cdot\nabla)\xi^k_{i,j}
+\sum_{k=1}^K\eta_k(\xi_{i,j}-\xi^k_{i,j})\cdot\partial_t\xi^k_{i,j}
\\&~~~
+\sum_{k=1}^K\xi_{i,j}\cdot\xi^k_{i,j}\partial_t\eta_k
+ O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
We proceed by inserting the equation for the time evolution of each local vector field $\xi^k_{i,j}$.
More precisely, by either employing the local equations \eqref{TransportEquationXiTwoPhase}
resp.\ \eqref{TransportEquationXiTriod} or making use of \eqref{GlobalEquationsBoundMinorityPhases},
we obtain together with \eqref{CompatibilityBound1} the estimate
\begin{align*}
\xi_{i,j}\cdot\partial_t\xi_{i,j}
&= -\sum_{k=1}^K\eta_k\xi_{i,j}\cdot(B^k\cdot\nabla)\xi^k_{i,j}
-\sum_{k=1}^K\eta_k(\xi_{i,j}-\xi^k_{i,j})\cdot(\nabla B^k)^\mathsf{T}\xi^k_{i,j}
\\&~~~
+\sum_{k=1}^K\xi_{i,j}\cdot\xi^k_{i,j}\partial_t\eta_k
+ O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
Using the compatibility bounds \eqref{CompatibilityBound5} resp.\
\eqref{CompatibilityBound1} and $\eta\leq O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)$,
we may compute
\begin{align*}
&\sum_{k=1}^K\eta_k(\xi_{i,j}-\xi^k_{i,j})\cdot(\nabla B^k)^\mathsf{T}\xi^k_{i,j}
\\
&=\sum_{k=1}^K\eta_k(\xi_{i,j}-\xi^k_{i,j})\cdot(\nabla B)^\mathsf{T}\xi^k_{i,j}
+ O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)
\\&
=(1-\eta)\xi_{i,j}\cdot(\nabla B)^\mathsf{T}\xi_{i,j}
-\xi_{i,j}\cdot(\nabla B)^\mathsf{T}\xi_{i,j}
+ O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1)
\\&
= O(\dist^2(\cdot,{\bar{I}}_{i,j})\wedge 1).
\end{align*}
Hence, together with the compatibility bound \eqref{CompatibilityBound6} we obtain
\begin{align*}
\xi_{i,j}\cdot\partial_t\xi_{i,j}
&= -\sum_{k=1}^K\eta_k\xi_{i,j}\cdot(B\cdot\nabla)\xi^k_{i,j}
+\sum_{k=1}^K\xi_{i,j}\cdot\xi^k_{i,j}\partial_t\eta_k
+ O(\dist^2(\cdot,\bar{I}_{i,j})\wedge 1).
\end{align*}
The remaining two sums may be rewritten as follows
\begin{align*}
&-\sum_{k=1}^K\eta_k\xi_{i,j}\cdot(B\cdot\nabla)\xi^k_{i,j}
+\sum_{k=1}^K\xi_{i,j}\cdot\xi^k_{i,j}\partial_t\eta_k
\\&
= -\xi_{i,j}\cdot(B\cdot\nabla)\xi_{i,j}
+\sum_{k=1}^K\xi_{i,j}\cdot(\xi^k_{i,j}{-}\xi_{i,j})
\big(\partial_t\eta_k{+}(B\cdot\nabla)\eta_k\big)
\\&~~~
-|\xi_{i,j}|^2\big(\partial_t\eta{+}(B\cdot\nabla)\eta\big).
\end{align*}
The last term in the latter identity is of required order because of \eqref{AdvectionEquationBulkLocalization}.
The term with the sum over the topological features may be treated by distinguishing between
two-phase interfaces $k\in\mathcal{K}^{\mathrm{2ph}}$ and triple junction $k\in\mathcal{K}^{\mathrm{3j}}$
as at the end of the proof of \eqref{TransportEquationXiGlobal}.
This concludes the proof of Lemma~\ref{LemmaGlobalCoercivityBounds}.
\end{proof}

Let us summarize our results from the previous sections to conclude with
a proof of the main result, Proposition~\ref{PropositionExistenceCalibration}.

\begin{proof}[Proof of Proposition~\ref{PropositionExistenceCalibration}]
Let $(\xi_{i,j})_{i\neq j}$ be the family of global vector fields
from Definition~\ref{DefinitionGlobalXi}.

Let $i,j \in \{1,\ldots,P\}$ with $i\neq j$ be fixed. The coercivity condition
\eqref{LengthControlXi} immediately follows from Lemma~\ref{lem:coercivityLenghtXi}.
The formula \eqref{Calibrations} follows from the corresponding local version \eqref{CalibrationLocalConstruction}
and the definition \eqref{GlobalXi}. Moreover, that $\xi_{i,j}(x,t)=\bar{\vec{n}}_{i,j}(x,t)$ holds true
for $t\in [0,T_{\mathrm{strong}})$ and $x\in {\bar{I}}_{i,j}(t)$
is a consequence of Property \textit{iii)} of Lemma~\ref{LemmaLocalConstructionsNetwork}
and that $(\eta_1,\ldots,\eta_K)$ is a partition of unity on the network of interfaces of the strong solution
(see Property \textit{i)} of Lemma~\ref{LemmaPartitionOfUnity}).

Finally, let $B$ be the global velocity field from Definition~\ref{DefinitionGlobalB}.
The validity of the equations \eqref{TransportEquationXi}, \eqref{LengthConservation}
and \eqref{Dissip} is the content of Lemma~\ref{LemmaGlobalCoercivityBounds}.
This concludes the proof of Proposition~\ref{PropositionExistenceCalibration}.
\end{proof}

\section*{Glossary of notation}

\begin{longtabu}{ll}
\endfirsthead
\endhead
\endfoot
\endlastfoot

$d\geq 2$ & ambient dimension
\vspace{2mm}
\\
$D$ & open set
\vspace{2mm}
\\
$\partial_t v$ & distributional partial derivative w.r.t.\ time
\\
&  of $v: D \times [0,T) \to \Rd$
\vspace{2mm}
\\
$\nabla v$ & distributional partial derivative w.r.t.\ space, $(\nabla v)_{i,j} = \partial_j v_i$
\vspace{2mm}
\\
$C_{\mathrm{cpt}}^\infty(D)$ & space of compactly
supported and infinitely\\
& differentiable functions on $D$
\vspace{2mm}
\\
$u \otimes v$ & tensor product of $u, v\in \Rd$, $(u \otimes v)_{i,j} = u_i v_j$
\vspace{2mm}
\\
$A : B$ & $\sum_{i,j} A_{ij}B_{ij}$, scalar product of tensors
\vspace{2mm}
\\
$\mathcal{L}^d$ & $d$-dimensional
Lebesgue measure
\vspace{2mm}
\\
$\mathcal{H}^k$ & $k$-dimensional Hausdorff measure on $\Rd$ for $k\in [0,d]$
\vspace{2mm}
\\
$L^p(\Omega,\mu)$ & Lebesgue space w.r.t.\ to a measure $\mu$ on $\Omega \subset \Rd$ for $p\in [1,\infty]$
\vspace{2mm}
\\
$L^p(D)$ & Lebesgue space w.r.t.\ Lebesgue measure
\vspace{2mm}
\\
$L^p(D;\Rd)$ & Lebesgue space for vector valued functions
\vspace{2mm}
\\
$L^p([0,T];X)$ & Bochner--Lebesgue space for a Banach space $X$ and $T \in (0,\infty)$
\vspace{2mm}
\\
$W^{k,p}(D)$ & Sobolev spaces with $p\in [1,\infty)$ and $k\in \mathbb{N}$
\vspace{2mm}
\\
$\|\cdot\|_{L^p_tW^{k,q}_x}$ & norm of $L^p([0,T];W^{k,q}(D))$ for $k\in \mathbb{N}$ and $q\in (1,\infty)$
\vspace{2mm}
\\
$L^p_tW^{k,q}_x$ & by abuse of notation, Sobolev space on the space-time
\\
& domain $\bigcup_{t\in [0,T]}D(t)\times\{t\}$
\vspace{2mm}
\\
$BV(D)$
& Functions of bounded variation \cite{AmbrosioFuscoPallara} on Lipschitz domain $D\subset \Rd$
\vspace{2mm}
\\
$\partial^* \Omega$ & reduced boundary of a set of finite perimeter $\Omega \subset D$
\vspace{2mm}
\\
$\smash{\vec{n}=-\frac{\nabla\chi_\Omega}{|\nabla\chi_\Omega|}}$ &  outward pointing unit normal vector field along $\partial^*\Omega$
\vspace{2mm}
\\
$\sdist(\cdot ,\partial \Omega)$ & signed distance function to $\partial \Omega$ with $\sdist(x,\partial \Omega)\leq 0$ for $x\in \Omega $
\vspace{2mm}
\\
$P \geq 2$ & number of phases
\vspace{2mm}
\\
$\Omega_i$ & region occupied by phase $i =1,\ldots,P$ in \emph{weak solutions}
\vspace{2mm}
\\
$\chi_i$ & characteristic function of $\Omega_i$
\vspace{2mm}
\\
$I_{i,j}$ & interface between phases $\Omega_i$ and $\Omega_j$
\vspace{2mm}
\\
$\vec{n}_{i,j}$ & unit normal vectors along $I_{i,j}$ pointing from phase $i$ to phase $j$
\vspace{2mm}
\\
$V_i$ & normal velocity of $I_{i,j}$ with $V_i>0$ for expanding $\Omega_i$, see \eqref{EvolutionPhasesBVSolution}
\vspace{2mm}
\\
$\bar \Omega_i$, $\bar \chi_i$, \ldots & corresponding quantities of the \emph{strong solution}
\vspace{2mm}
\\
${\bar{\vec{H}}}_{i,j}$ & mean curvature vector of ${\bar{I}}_{i,j}$
\vspace{2mm}
\\
${\bar{H}}_{i,j}$ & scalar mean curvature of ${\bar{I}}_{i,j}$ given by $-\nabla^{\mathrm{tan}}\cdot\bar{\vec{n}}_{i,j}=-\Delta s_{i,j}$
\vspace{2mm}
\\
$s_{i,j}$ & signed distance function to ${\bar{I}}_{i,j}$ with $\nabla s_{i,j} = \bar{\vec{n}}_{i,j}$
\vspace{2mm}
\\
$\smash{J= \big(\begin{smallmatrix} 0 & -1 \\ 1 & 0	\end{smallmatrix}\big)}$ & counter-clockwise rotation by $90^\circ$
\vspace{2mm}
\\
$\bar{\tau}_{i,j}$ & tangent vector along ${\bar{I}}_{i,j}$ given by $ J^{-1}\bar{\vec{n}}_{i,j}$
\vspace{2mm}
\\
$O(\cdot)$ & Landau symbol, implicit constant only depends on strong solution
\end{longtabu}

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redpajama_set_name: RedPajamaArXiv