\begin{document}

\title{The influence of boundary conditions on the contact problem \\Â in a 3D Navier-Stokes Flow}
\author{David Gerard-Varet\footnote{Institut de Math\'ematiques de Jussieu, CNRS UMR 7586, Universit\'e Paris 7, 175 rue du Chevaleret, 75013 Paris. Mail: \texttt{gerard-varet@math.jussieu.fr}},\, 
Matthieu Hillairet\footnote{Ceremade, CNRS UMR 7534, Univerisit\'e Paris Dauphine, Place du Mar\'echal de Lattre de Tassigny, 75775 Paris Cedex 16  
 and Institut Camille Jordan, CNRS UMR 5208, Universit\'e Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbane Cedex, Mail: \texttt{hillairet@ceremade.dauphine.fr}}\,   
 and Chao Wang\footnote{Institut de Math\'ematiques de Jussieu, CNRS UMR 7586, Universit\'e Paris 7, 175 rue du Chevaleret, 75013 Paris. Mail: \texttt{wangc@math.jussieu.fr}} 
 }

\date{\today}
\maketitle

\begin{abstract}
We consider the free fall of a sphere above a wall in a viscous incompressible fluid. We investigate the influence of
boundary conditions on the finite-time occurrence of contact between the sphere and the wall. We prove that
slip boundary conditions enable to circumvent the "no-collision" paradox associated with no-slip boundary conditions.
We also examine the case of mixed boundary conditions.
\end{abstract}

\section{Introduction}
Understanding the close-range dynamics of particles inside a fluid is crucial to several fields of application, such as  rheology, sedimentation or slurry erosion. A related problem is to compute the drag induced by nearby solid  bodies: it is important to lubrication theory, or to the control of microswimmers, among others. In the case of a viscous incompressible fluid, a standard way to model the fluid-solid interaction is through the coupling of the Navier-Stokes equations for the fluid part and the conservation of momentum for the bodies.  However, such model reveals unrealistic features: in particular, it can overvalue the drag force induced on  bodies that are close to collision.  Consider for instance the free fall of a rigid sphere over a wall, and assume standard no-slip conditions at the solid surface of the sphere and the wall. Then, the model predicts that no collision is possible between the sphere and the wall, no matter their relative density and the viscosity of the fluid.This is a "no-collision paradox": it has been recognized at first for reduced  models (Stokes approximation, see \cite{Cooley&Oneill69,Cox74}), and more recently for the  full constant-density incompressible Navier Stokes system \cite{DGVH10,Hillairet07b,HillairetTakahashi09}. These results show the need for a more accurate model of the fluid behavior in the "small-distance" regime.  Several refinements were proposed in the past \cite{BarnockyDavis89,davisetal86,KytomaaSchmid92}, see the introduction of \cite{JosephPhD} for a review. In this paper, we investigate the influence of boundary conditions at the fluid-solid interfaces.

Boundary conditions are constitutive equations that can be derived theoretically or experimentally. A popular general type of boundary conditions was proposed by Navier
in  the early time of analytical fluid dynamics. This Navier law states that, on the one hand, the normal component of the velocity is continuous
at the fluid/solid interface (impermeability condition) and that, on the other hand, the amount of slip in the tangential part of the velocity is proportional to the
tangential part of the normal stress exerted by the fluid on the boundary. The proportionality coefficient is called "slip length". Assuming the slip length vanishes (\emph{i.e.}, assuming no-slip boundary conditions) suffices to make theoretical and experimental results to coincide in numerous cases. However, developments such as the "no-collision paradox" lead to question this assumption  \cite{Hocking}.
We refer the reader to \cite{Bocquet,Lauga} for a further description of experimental and theoretical validations of the Navier boundary conditions with slip.
{\em We propose here new insights on the contact-problem, by considering the effect of Navier boundary conditions. We shall study  this effect in  the context of the full Navier-Stokes system, which is, up to our knowledge, new.}

We consider a  simplified case where only one homogeneous solid is moving in the fluid.
We denote $S(t)\subseteq\mathbb{R}^3, F(t)\subseteq\mathbb{R}^3$ the solid
and the fluid domains at time $t$, and $\Omega\triangleq \overline{S(t)}\cup F(t)$ the total domain. We assume that
the fluid is governed by the constant-density incompressible Navier-Stokes equations :
\begin{eqnarray}\rho_F(\partiala_t u_F+u_F\cdot\nablabla u_F)-\mu_F\Delta u_F &=& -\nablabla p_F-\rho_Fge_3, \partialhantom{0}  t>0, \quad x\in F(t), \frak lambdabel{F1}\\
\mbox{div} \ u_F&=& 0,   \partialhantom{-\nablabla p_F-\rho_Fge_3} t>0,\quad x\in F(t), \frak lambdabel{F2}
\end{eqnarray}
where $u_F$ and $p_F$ stand respectively for the velocity and internal pressure, $\rho_F$ the density,
$\mu_F$ the viscosity in the fluid. We emphasize that we consider here the influence of gravity whose direction
is the third vector of the basis $(e_1,e_2,e_3),$ \emph{i.e.}, the "vertical" unit vector.
The solid motion is governed by Newton laws.
Denoting $x_S(t)\in \mathbb{R}^3$ the position of the center of mass of the solid
at time $t$, $U_S(t)\in \mathbb{R}^3$ its velocity and $\omega_S(t)\in \mathbb{R}^3$ the angular velocity, the Newton laws read
\begin{eqnarray}
m_S\dfrac{d}{dt}U_S(t)&=&-\displaystyle{\int_{\partiala S(t)}} (2\mu_FD(u_F)-p_FI)n d\sigma- m_Sge_3, \frak lambdabel{S1}\\
\dfrac{d}{dt}(J_S \omega_S(t))&=&-\displaystyle{\int_{\partiala S(t)}} (x-x_S(t)) \times (2\mu_FD(u_F)-p_FI)n   d\sigma. \frak lambdabel{S2}
\end{eqnarray}
Here $\rho_S$ and $m_S\triangleq \rho_S|S(0)|$ are the density and the mass of the solid respectively. Symbol $n$  stands for the unit normal vector to $\partialartial S(t)$ pointing outside the fluid domain. The 3x3 inertia matrix $J_S(t)$ is defined by
\begin{eqnarray}o
J_S(t)\triangleq \rho_S\int_{S(t)}(|x-x_S(t)|^2I_d-(x-x_S(t))\otimes(x-x_S(t)))dx.
\end{eqnarray}o
We emphasize that the solid is homogeneous, so that  no angular momentum  is induced by gravity in \eqref{S2}.
Given $(U_S,\omega_S,x_S),$ the solid velocity-field $u_S$ at each point $x \in S(t)$ reads
\begin{equation} \frak lambdabel{eq_solidvel}
u_S(t,x)\triangleq U_S(t)+\omega_S\times(x-x_S(t)).
\end{equation}
In this paper, we complete the system with Navier boundary conditions :
\begin{eqnarray}
(u_F-u_S)\cdot n|_{\partiala S(t)}=0,&&\frak hspace{-42pt} \partialhantom{u_F\times n|_{\partiala\Omega}}(u_F-u_S)\times n|_{\partiala S(t)}=-2\beta_S(D(u_F)n)\times n, \frak lambdabel{bc1}\\
u_F \cdot n|_{\partiala\Omega}=0,   && \frak hspace{-42pt}\partialhantom{(u_F-u_S)\times n|_{\partiala S(t)}}u_F\times n|_{\partiala\Omega}=-2\beta_{\Omega}(D(u_F)n)\times n, \frak lambdabel{bc2}
\end{eqnarray}
where $\beta_S, \beta_\Omega\geq 0$ are the slip lengths at the fluid-solid and fluid-container interface respectively.
We remind that assuming that $\beta_S$ or $\beta_{\Omega}$ vanishes means assuming no-slip boundary
conditions at the corresponding interface.

System \eqref{F1}--\eqref{bc2} is completed with initial conditions :
\begin{eqnarray}
S(0) = S_0, && u_F(0,x) = u_{F,0}(x), \quad x \in \Omega \setminus \bar{S}_0 \, \frak lambdabel{ic1}\\
U(0) = U_0, && \omega(0) = \omega_0\,. \frak lambdabel{ic2}
\end{eqnarray}
Equation \eqref{ic2} may also be written :
\addtocounter{equation}{-1}
\begin{equation}
u_S(0,x) = u_{S,0}(x), \quad  x \in S_0\,.
\end{equation}
 where $u_{S,0}$ is computed with respect to $U_0,\omega_0$ and $S_0$ {\em via} \eqref{eq_solidvel}.
When $S_0 \subset \Omega$ and  $u_{F,0}$ has finite energy, namely $u_{F,0} \in L^2(\Omega \setminus \bar{S}_0),$ standard  computations
show that a reasonable solution to \eqref{F1}--\eqref{ic2} should satisfy the energy estimate :
\begin{multline} \frak lambdabel{nrj_intro}
\dfrac{1}{2} \frak left[ \int_{F(t)} \rho_F |u_F(t)|^2 + \int_{S(t)} \rho_S |u_S(t)|^2 \right]   \\
+  \mu_F \frak left(  2  \int_{0}^t \int_{F(t)} |D(u_F(t))|^2 + \dfrac{1}{\beta_{\Omega}} \int_{0}^t \int_{\partialartial \Omega} |u_F \times n|^2 {\rm d}\sigma  + \dfrac{1}{\beta_S} \int_0^t \int_{\partialartial S(t)} |(u_F - u_S) \times n |^2 {\rm d}\sigma \right)
\\
\frak leq
- \int_0^t \int_{F(t)} \rho_F g e_3 \cdot u_F - \int_0^t \int_{S(t)} \rho_S g e_3 \cdot u_S +\dfrac{1}{2} \int_{F_0} \rho_F |u_{F,0}|^2 +\dfrac{1}{2}\int_{S_0} \rho_S |u_S(0)|^2\,.
\end{multline}
where $F_0 \triangleq \Omega \setminus \bar{S}_0.$
When $\beta_S = 0$ or $\beta_\Omega = 0$,  the corresponding boundary term vanishes.

Roughly, with regards to the classical theory of  the  Navier-Stokes equation in a fixed domain, one can expect two kinds of solutions for \eqref{F1}--\eqref{ic2}:
\begin{itemize}
\item Strong solutions, locally in time: they should be unique, with possible blow-up  prior to collision in 3D.
\item Weak solutions: they should exist at least up to collision, and be possibly non-unique.
\end{itemize}
In  the no-slip case  ($\beta_S= \beta_{\Omega} =0$), both kinds of solutions have been built in various contexts: 2D/3D problems, bounded/unbounded
container, smooth/singular shape of the solid bodies. Existence of strong solutions locally in time (prior to collision) is now well-known, see \cite{Desjardins&Esteban99,GaldiSilvestre05,DGVH10,Grandmont&Maday98,Takahashi03bis,Takahashi03}. The existence of weak solutions up to collision has been studied in
\cite{Conca&al00,CumsilleTakahashi,Desjardins&Esteban00,Gunzburger&Lee&Seregin00,Hoffmann&Starovoitov99,Hoffmann&Starovoitov00,Serre87,Takahashi&Tucsnak04}. Uniqueness of weak solutions up to collision in 2D is established in the recent paper \cite{GlassSueurpp}.  Finally , extension  of weak solutions after contact is given  in \cite{SanMartin&al02} for 2D problems
and \cite{Feireisl03} for 3D problems.

In the case of slip conditions, when $\beta_S >0$ and $\beta_{\Omega} >0$, much less in known.  It is partly because genuine new mathematical difficulties arise: for instance, at the level of weak solutions, discontinuity of the tangential velocity at the fluid/solid interface forbids global $H^1$ bound on the extended velocity-field $u$ as defined by:
\begin{equation} \frak lambdabel{intro_trick}
u = u_S  \text{ in $S(t)$}\,, \qquad u= u_F \text{ in $F(t)$}.
\end{equation}
Such global bound is a key point in the treatment of the Dirichlet case. Hence, to investigate the contact problem for system  \eqref{F1}--\eqref{ic2}, we shall rely on the recent paper \cite{DGVHpp}, which  shows the existence of weak solutions up to collision. Both the definition of weak solutions and the existence result given in  \cite{DGVHpp} will be recalled in the next section.  Let us also mention the work  \cite{SueurPlanaspp}, related to a single body without container.

{\em Considering a special class of weak solutions, we will show that slip conditions allow collision in finite time}. Hence, taking into account slight slip at the solid boundaries allows to clear the no-collision paradox. More precisely, we shall consider configurations obeying the following assumptions:
\begin{enumerate}
\item[A1.] The solid $S(t) \sim S_0$ is a ball of radius $1$.
\item[A2.] $\Omega$ is a smooth convex domain of $ \{ x_3 > 0\}$,  flat near:
$$\partiala \Omega \supset\{ x_3 = 0, |x| < 2 \delta \}  \: \mbox{ for some } \: \delta \in (0,\frac{1}{4}).$$
\item[A3.] $S(t)$ is axisymmetric with respect to $\{x_1=x_2=0\}$.
\item[A4.] The only possible contact  between the solid  and the container is at $x=0$. More precisely,
$$ dist\frak left(S(t), \partiala \Omega \setminus \{ x_3 = 0, |x| < \delta \} \right) \ge d_\delta > 0. $$
\end{enumerate}
Parameters $\delta$ and $d_\delta$ will be involved in the proof. Let us comment on these  assumptions. {\em Their only goal is to  ensure that  the geometry of the potential contact zone does not change with time.} Namely, it is an aperture between a sphere and a plane, only varying with the distance $h(t)$ between the south pole of $S(t)$ and $x=0$. The whole point is then to determine if $h(t)$ can vanish in finite time or not. This simplifies greatly our computations. However, as will be clear from our proof, our result is genuinely local in space time: it  is independent from the global characteristics of the solid/container geometry.

Note that  A1-A2 are just assumptions on $S_0$ and  $\Omega$. If $S_0 \subset \subset \Omega$, and  if the initial data $(u_{F,0},u_{S,0})$ has finite energy, article \cite{DGVHpp} ensures the existence of a weak solution up to contact between $S(t)$ and $\partialartial \Omega$ (see next section). Assumption A3 states that the axisymmetry of the solid is preserved with time. Actually, it will be automatically satisfied if $S_0$, $\Omega$ and the initial velocities are axisymmetric.  More precisely, in this case, one can adapt the construction of weak solutions  from \cite{DGVHpp} by including  the symmetry constraint, and obtain directly an axisymmetric weak solution. Finally, assumption A4  guarantees that the sphere remains away from the top and lateral boundaries. Again, this can be ensured by a proper choice of the container and the initial data. As regards lateral boundaries, it is enough by assumption A3 that they are at distance $> 1$ to the  vertical axis $\{x_1 = x_2 = 0\}$. Also,  there are several ways to avoid merging of the solid body and the top boundary. One can for instance assume that the solid is heavier than the fluid ($\rho_S > \rho_F$), and that the initial kinetic energy of the fluid and the solid is small enough. As the total energy of weak solutions is non-increasing, this means that the increase of the potential energy through time can only be small: hence, the solid will not rise too much, and so will remain away from the top boundary. Again, we insist that our method of proof could extend to more general settings.

We can now state our main results:
\begin{theorem}[Slip Case] \frak lambdabel{thm_slip}
Assume  $\beta_S, \beta_{\Omega} > 0$ and  $\rho_S > \rho_F$. For any weak solution $(S,u)$  (see Definition  \ref{def:ws_s}) satisfying A1-A4, the solid body $S$  touches  $\partiala \Omega$ in finite time.
\end{theorem}

\begin{theorem}[Mixed Case] \frak lambdabel{thm_mixed}
Let $\beta_S$ or $\beta_{\Omega}$ vanish. Any weak solution $(S,u)$  (see Definition  \ref{def:ws_m}) satisfying A1-A4 is global. In particular, the solid body $S$ never touches
$\partialartial \Omega.$
\end{theorem}
We recall that the full no-slip case (when $\beta_S = \beta_{\Omega} = 0$) was already treated in \cite{HillairetTakahashi09}.
Theorems \ref{thm_slip} and \ref{thm_mixed} show that one possible way  to obtain realistic contact, and to circumvent the no-collision paradox,
 is to introduce slip in the boundary conditions {\em on all solid boundaries of the fluid domain}.

We conclude with  a few hints on the proof of Theorem \ref{thm_slip} (ideas for Theorem  \ref{thm_mixed} are similar). The proof elaborates on the paper \cite{DGVH12}, devoted to a simplified linear system:  one  takes there $\Omega = \mathop{\bf R\kern 0pt}\nolimits^3_+$, while  the fluid outside the sphere $S(t)$ is governed by a steady Stokes flow (with $\mu_F = 1$ for simplicity):
 $$ - \Delta u_F = -\nablabla p_F-\rho_Fge_3, \quad \nabla \cdot u_F = 0, \quad x \in F(t).  $$
In this simplified setting, the fluid and solid domains $F(t) = F_{h(t)}$ and $S(t) = S_h(t)$  are characterized by the distance $h(t)$ beween the sphere and the plane wall. Then, it is easily seen that $u_F(t) = h'(t) w_{h(t)}$ where $w_h$ is the solution of a normalized Stokes equation, set in the domain $F_h$  "frozen"  at distance $h$:
 \begin{equation} \frak lambdabel{wh}
 \frak left\{
\begin{aligned}
  -\Delta w_h +  \nablabla p_h   = 0, & \quad \nabla \cdot w_h = 0, \quad x \in F_h,  \quad (w_h-e_3)\cdot n\vert_{\partiala S_h}=0,  \quad w_h \cdot n|_{\partiala\Omega}=0, \\
 (w_h-e_3)\times n\vert_{\partiala S_h} &=-2\beta_S(D(w_h)n)\times n, \\
 w_h \times n|_{\partiala\Omega} & =-2\beta_{\Omega}(D(w_h)n)\times n.
 \end{aligned}
 \right.
 \end{equation}
 We refer to \cite{DGVH12} for all details. Eventually, the dynamics reduces to  an ODE of the type
  $$ h''(t) = - h'(t) {\cal D}(h(t)) + \frac{\rho_S - \rho_F}{\rho_S} g $$
where the {\em drag term}  ${\cal D}(h)$ is the energy of $w_h$:
$$ {\cal D}(h) \: =  \: {\cal E}_h(w_h), \quad {\cal  E}_h(u) \: :=  \:  \int_{F_h} | \nabla u|^2 + \frak left( \frac{1}{\beta_S} + 1\right) \int_{\partiala S_h} | (u - e_3) \times n |^2  + \frac{1}{\beta_\Omega}  \int_{\partiala \Omega} | u \times n |^2. $$
 For this reduced model, the contact problem resumes to the computation of the drag ${\cal D}_h$ in the limit $h \rightarrow 0$. The difficulty is that  there is no simple formula for the solution $w_h$ of the Stokes system in $F_h$.

 

 A general approach to this problem was recently proposed in \cite{DGVH12}. It works for no-slip conditions, slip ones, and for more general geometries. The starting idea is to use a variational characterization of the drag: one can identify  ${\cal D}(h)$ as the minimizer of an energy ${\cal E}_h$, over an admissible set of fields ${\cal A}_h$:
 $$ {\cal D}(h) \: = \inf_{u \in {\cal A}_h} {\cal E}_h(u) = {\cal E}_h(w_h). $$
 Then, instead of computing the true minimizer, the point is to find an appropriate relaxed minimization problem, for which  the minimizer  can be easily computed, and such that the corresponding minimum is close to the exact one asymptotically in $h$. The choice of such relaxed problem is developed in various situations in  \cite{DGVH12}.
Roughly,  in the case of slip conditions, one obtains  that $c |\frak ln (h)| \: \frak le \:  D(h) \: \frak le \: C |\frak ln(h)|$, in agreement with older computations of Hocking \cite{Hocking}. Hence, the drag is weak enough to allow for collision, within this linear Stokes approximation. See also \cite{HLS11} for numerical insight.

 

 In the present paper, we manage to use ideas of the  linear study, in the context of the nonlinear Navier-Stokes flow. At a formal level, the first idea is to multiply the momentum equation by $w(t,x) = w_{h(t)}(x)$, $w_h$ solution of \eqref{wh} (where this time $\Omega$ is the container). After formal  manipulations, one obtains
 \begin{equation} \frak lambdabel{doth}
  h''(t)  \: = \:  - h' {\cal D}(h)   \: + \frac{\rho_S - \rho_F}{\rho_S} g \: + \: \mbox{remainder}
  \end{equation}
  where the remainder term involves the test function $w$ and the convective derivative of $u$. However, the Stokes solution $w_h$ is not known explicitly. {\em Hence, the main point in our proof is to use another test function, based on the relaxed minimizer introduced in \cite{DGVH12}. Then, one must show that for such test function, the remainder term is indeed small, that is controlled by the gravity when the solid is heavier than the fluid}. Let us stress that several difficulties arise in this process.  For instance, the relaxed minimizer introduced in the linear study is not accurate enough, so that we must improve it. Moreover, due to the slip conditions, the natural quantity is the symmetric gradient of the weak solution, not the full gradient. This makes the derivation of the various bounds more difficult than in the no-slip case.

The outline of the paper is as follows. In the next section,  we introduce a suitable notion of weak solutions for system \eqref{F1}--\eqref{ic2},  in the slip case and in the mixed case. We recall the existence theorems up to collision available for such weak solutions. in the slip case and in the mixed case. The two last sections are devoted to the proofs of the two theorems.

\section{On the definition of weak solutions}

We introduce here a definition of weak solutions for \eqref{F1}--\eqref{ic1} in  two cases: slip boundary conditions at all boundaries, mixed slip/no-slip
boundary conditions. We mostly follow \cite{DGVHpp}.  Given $\Omega$ a Lipschitz domain, we set:
\begin{eqnarray*}
\mathcal{D}_{\sigma}(\Omega)& \triangleq&  \{ \varphi\in C^\infty_c(\Omega),\quad \mbox{div} \, \varphi=0 \},
     \qquad \partialhantom{1121}\mathcal{D}_\sigma(\overline\Omega)\triangleq\{ \varphi|_{\Omega}, \: \varphi\in\mathcal{D}_\sigma(\mathbb{R}^3) \},\\[4pt]
L^2_{\sigma}(\Omega) &\triangleq& \text{\rm the closure of } \mathcal{D}_{\sigma}(\Omega) \textrm{ in } L^2(\Omega),
   \qquad H^1_{\sigma}(\Omega) \triangleq H^1(\Omega)\cap L^2_\sigma(\Omega),\\[4pt]
 H^1_{\sigma}(\overline\Omega) &\triangleq & \textrm{ the closure of } \mathcal{D}_{\sigma}(\overline\Omega) \textrm{ in }  H^1(\Omega),\\[4pt]
 \mathcal{R} &\triangleq &\{\varphi_s,\quad \varphi_s=V+\omega\times x,\quad\textrm{for some}\quad V\in\mathbb{R}^3, \omega\in \mathbb{R}^3 \},
\end{eqnarray*}

\subsection{The case of slip boundary conditions: $\beta_S >0$ and $\beta_{\Omega} >0$.}
We consider here that the slip lengths $\beta_S$ and $\beta_{\Omega}$ do not vanish \emph{i.e.}, that the fluid slips on both boundaries.
The corresponding Cauchy problem is studied in \cite{DGVHpp}, where the following definition of weak solution is given:
\begin{definition} \frak lambdabel{def:ws_s}
Let $\Omega$ and $S_0\Subset\Omega$ be two Lipschitz bounded domains of $\mathbb{R}^3$. Let $u_{F,0}\in L^2_{\sigma}(\Omega), u_{S,0}\in \mathcal{R}$ such that
$u_{F,0}\cdot n=u_{S,0}\cdot n$ on $\partiala S_0$. Let $T > 0$, or $T =\infty$.

 A weak solution of (\ref{F1})--(\ref{ic2}) on $[0,T)$ is a pair $(S, u)$ satisfying

 

 \noindentindent
1) $S(t)\Subset \Omega$ is a bounded domain of $\mathbb{R}^3$ for all $t\in[0,T)$ such that
\begin{eqnarray}o
\chi_S(t,x)\triangleq 1_{S(t)}(x)\in L^\infty((0,T)\times\Omega).
\end{eqnarray}o
2) $u$ belongs to the space
\begin{multline*}
\mathcal{S}_T\triangleq \{ u\in L^\infty(0,T;L^2_\sigma(\Omega)),  \textrm{ there exists } u_F \in L^2_{loc}([0,T); H^1_\sigma(\Omega)), \: u_S\in L^\infty(0,T; \mathcal{R})  \\
  \partialhantom{1qdfZERTYUT}  \textrm{ such that }  \:  u(t,\cdot)=u_F(t,\cdot) \textrm{ on } F(t),    \quad u(t,\cdot)=u_S(t,\cdot) \textrm{ on } S(t),\, \textrm{for all } t\in[0,T)\}
\end{multline*}
where $F(t) \triangleq \Omega \setminus \overline{S(t)}$. \\
3) For all $\varphi$ belonging to the space
\begin{multline*}
\mathcal{T}^{s}_{T} \triangleq \{\varphi\in C([0,T];L^2_{\sigma}(\Omega)),
\textrm{ there exists } \varphi_F\in \mathcal{D}([0,T);\mathcal{D}_{\sigma}(\overline\Omega)),
   \varphi_S\in \mathcal{D}([0,T);\mathcal{R})\\
  \partialhantom{1qdfZERTYUT}\textrm{ such that } \: \varphi(t,\cdot)=\varphi_F(t,\cdot) \, \textrm{ on }\, F(t),  \quad \varphi(t,\cdot)=\varphi_S(t,\cdot)\,\textrm{ on }\, S(t),\, \textrm{for all } t\in[0,T]\}\,
\end{multline*}
there holds
\begin{eqnarray}\frak lambdabel{ws_s}
&&-\int_0^T\int_{F(t)}\rho_F u_F\cdot\partiala_t\varphi_F - \int_0^T\int_{S(t)}\rho_S u_S\cdot\partiala_t\varphi_S - \int_0^T\int_{F(t)}\rho_Fu_F\otimes u_F: \nablabla \varphi_F
 \noindentnumber\\
&& + \f{\mu_F}{\beta_\Omega}\int_0^T\int_{\partiala\Omega}(u_F\times n)\cdot(\varphi_F\times n)
+ \f{\mu_F}{\beta_S}\int_0^T\int_{\partiala S(t)}((u_F - u_S) \times n)\cdot ((\varphi_F - \varphi_S)\times n)  \noindentnumber\\
&&+  \int_0^T\int_{F(t)}2\mu_FD(u_F):D(\varphi_F)  \quad = \quad -\int_0^T\int_{F(t)}\rho_F g e_3 \varphi_F\noindentnumber -\int_0^T\int_{S(t)}\rho_S g e_3 \varphi_S\noindentnumber\\
&& + \int_{F_0}\rho_F u_{F,0}\varphi_F(0,\cdot) + \int_{S_0}\rho_S u_{S,0}\varphi_S(0,\cdot),
\end{eqnarray}
4) For all $\partialsi \in \mathcal{D}([0,T);\mathcal{D}(\overline\Omega))$, we have
\begin{eqnarray}o
-\int^T_0\int_{S(t)}\partiala_t\partialsi-\int^T_0\int_{S(t)}u_S\cdot \nablabla \partialsi=\int_{S_0}\partialsi(0,\cdot),
\end{eqnarray}o
5) We have the following energy inequality for a.a. $t \in (0,T)$ :
\begin{multline} \frak lambdabel{nrj_est_s}
\dfrac{1}{2} \frak left[ \int_{F(t)} \rho_F |u_F(t)|^2 + \int_{S(t)} \rho_S |u_S(t)|^2 \right] \\
+  \mu_F \frak left(  2  \int_{0}^t \int_{F(t)} |D(u_F(t))|^2 + \dfrac{1}{\beta_{\Omega}} \int_{0}^t \int_{\partialartial \Omega} |u_F \times n|^2 {\rm d}\sigma
+\dfrac{1}{\beta_S} \int_0^t \int_{\partialartial S(t)} |(u_F - u_S) \times n |^2 {\rm d}\sigma \right) \\
\frak leq
- \int_0^t \int_{F(t)} \rho_F g e_3 \cdot u_F - \int_0^t \int_{S(t)} \rho_S g e_3 \cdot u_S + \dfrac{1}{2} \int_{F(0)} \rho_F |u_{F,0}|^2 + \dfrac{1}{2} \int_{S(0)} \rho_S |u_S(0)|^2\,.
\end{multline}
\end{definition}
We refer to \cite{DGVHpp}  for a detailed discussion of this definition.  Note that it is restricted to solutions up to collision, through the condition $S(t) \Subset \Omega$. We recall that item {\em 3)} is the identity one gets by multiplying \eqref{F1} by a divergence-free test-function $\varphi_F$ and then combining
with \eqref{S1}--\eqref{S2}. Item {\em 4)} is the weak form of the transport equation satisfied
by $\chi_S$, the indicator function of $S(t)$ as defined in {\em 1)}. Finally, {\em 2)} summarizes the regularity that is implied by the energy estimate \eqref{nrj_est_s}. Note that $u_F$ and $u_S$ share the same normal component, so that the $L^\infty L^2$ bound on $u_F$ and $u_S$ yields a global $L^\infty L^2_\sigma(\Omega)$ on  the extended velocity field $u$ (see \eqref{intro_trick} for the definition of $u$).
Moreover, the $L^\infty L^2$ bound on $u_F$ and $L^2 L^2$ bound on  $D(u_F)$ imply an $L^2_{loc}H^1$ bound, through Korn inequality (that is valid as long as the solid is away from the boundary of the container).

From \cite{DGVHpp}, there is a weak solution for the system up to a contact between $S(t)$ and $\partialartial \Omega$:
\begin{theorem}\frak lambdabel{thm:ws_s}
Let  $\Omega$ and $S_0 \Subset \Omega$ two $C^{1,1}$ bounded domain of $\mathbb{R}^3$.
Given $(\beta_S,\beta_{\Omega}) \in (0,\infty)^2,$ and initial data $u_{F,0}\in L^2_{\sigma}(\Omega), u_{S,0}\in \mathcal{R}$ satisfying:
$$u_{F,0}\cdot n= u_{S,0}\cdot n,\quad \textrm{on} \quad \partiala S_{0},$$
there exists  $T \in \mathop{\bf R\kern 0pt}\nolimits^*_+ \cup \{\infty\}$ and a weak solution of (\ref{F1})-(\ref{ic2}) on $[0,T)$. Moreover, such  weak solution exists up to collision, that is either we can take $T=\infty$, or we can take $T \in \mathop{\bf R\kern 0pt}\nolimits_+^*$ in such a way that
\begin{eqnarray}o
S(t)\Subset \Omega,\quad\textrm{for all}\quad t\in[0,T),\quad \textrm{and}\quad \frak lim_{t\rightarrow T^{-}}\textrm{dist}(\partiala S(t),\partiala\Omega)=0.
\end{eqnarray}o
\end{theorem}
\noindentindent The aim of {Theorem \ref{thm_slip}} is to prove that $T < \infty$ under assumption A1-A4.

\subsection{The mixed case: $\beta_S = 0$ and $\beta_{\Omega} >0$.}
We proceed with the case where slip is imposed on one of the boundaries only. For simplicity, we consider that no-slip boundary conditions are imposed on $\partialartial S(t)$
\emph{i.e.} $\beta_S = 0$ and $\beta_{\Omega} >0.$ To our knowledge, this particular system has not been  treated explicitly in previous studies. Nevertheless, as slip is allowed on the exterior boundary of $F(t)$ only, it is straightforward to adapt the method of \cite{Desjardins&Esteban00} or \cite{Hoffmann&Starovoitov00}, for instance, to tackle a Cauchy theory for weak solutions before contact. In particular, the extension trick \eqref{intro_trick} yields a sufficiently smooth velocity-field in this case  so that we can reduce the full system to a global weak formulation on this  extended velocity field. This reads as follows:
\begin{definition}   \frak lambdabel{def:ws_m}
Let $\Omega$ and $S_0\subset\Omega$ be two Lipschitz bounded domains of $\mathbb{R}^3$. Let $u_{F,0}\in L^2_{\sigma}(\Omega), u_{S,0}\in \mathcal{R}$ such that
$u_{F,0}\cdot n=u_{S,0}\cdot n$ on $\partiala S_0$. A weak solution of (\ref{F1})--(\ref{ic2}) is a pair $(S, u)$ satisfying

 \noindentindent
1) $S(t)\Subset \Omega$ is a bounded domain of $\mathbb{R}^3$ for all $t\in[0,T)$ such that
\begin{eqnarray}o
\chi_S(t,x)\triangleq 1_{S(t)}(x)\in L^\infty((0,T)\times\Omega).
\end{eqnarray}o
2) $u \in L^{\infty}(0,T;L^2_{\sigma}(\Omega)) \cap L^2(0,T;H^1_{\sigma}(\Omega))$ satisfies
$$
u(t,\cdot)_{|_{S(t)}} \in \mathcal R \quad \text{ for a.a. $t \in (0,T)\,.$}
$$
3) For all $\varphi$ belonging to
$$
\mathcal{T}^{m}_{T} \triangleq  \{\varphi\in \mathcal D([0,T) \times \bar \Omega), \quad \varphi(t,\cdot)_{|_{S(t)}} \in \mathcal R \text{ for all $t \in [0,T)$}\}
$$
there holds
\begin{multline}\frak lambdabel{ws_m}
-\int_0^T\int_{\Omega}\rho \frak left( u \partiala_t\varphi  +  u \otimes u : D(\varphi) \right)  +\int_0^T\int_{\Omega}2\mu_F D(u):D(\varphi) \\
 + \f{1}{\beta_\Omega}\int_0^T\int_{\partiala\Omega}(u \times n)\cdot(\varphi \times n)
  =-\int_0^T\int_{\Omega}\rho  g e_3 \varphi -\int_{\Omega}\rho_0 u_0 \varphi(0,\cdot),
\end{multline}
where
\begin{eqnarray*}
\rho(t,x) &=& \rho_S \, 1_{S(t)}(x) + \rho_F 1_{F(t)}(x), \quad \text{ for $(t,x) \in (0,T) \times \Omega,$}\\
\rho_0(x)& = &\rho_S \, 1_{S_0}(x) + \rho_F 1_{F_0}(x),  \quad \partialhantom{.22} \text{for $x \in \Omega,$} \\
 u_{0}(x) &=& u_{S,0} \, 1_{S_0}(x) + u_{F,0} 1_{F_0}(x),  \quad \partialhantom{.22} \text{for $x \in \Omega,$}
\end{eqnarray*}
4) For all $\partialsi \in \mathcal{D}([0,T);\mathcal{D}(\overline\Omega))$, we have
\begin{eqnarray}o
-\int^T_0\int_{S(t)}\partiala_t\partialsi -\int^T_0\int_{S(t)}u \cdot \nablabla \partialsi=\int_{S_0}\partialsi(0,\cdot),
\end{eqnarray}o
5) We have the following energy inequality for a.a. $t \in (0,T)$ :
\begin{multline} \frak lambdabel{nrj_est_m}
\dfrac{1}{2}  \int_{\Omega} \rho |u(t)|^2 + 2 \mu_F \int_{0}^t \int_{\Omega} |D(u)|^2  + \dfrac{1}{\beta_{\Omega}} \int_{0}^t \int_{\partialartial \Omega} |u  \times n|^2 {\rm d}\sigma  \\
\frak leq
- \int_0^t \int_{F(t)} \rho g e_3  u  + \dfrac{1}{2}\int_{\Omega} \rho(0,\cdot) |u_{0}|^2 \,.
\end{multline}
\end{definition}

Following the method of proof of \cite{Desjardins&Esteban00}, one obtains:

\begin{theorem}\frak lambdabel{thm:ws_m}
Let  $\Omega$ and $S_0\Subset \Omega$ two $C^{1,1}$ bounded domain of $\mathbb{R}^3$.
Given $\beta_{\Omega} \in (0,\infty)$ and initial data $u_{F,0}\in L^2_{\sigma}(\Omega), u_{S,0}\in \mathcal{R}$ satisfying
$$  u_{F,0}\cdot n= u_{S,0}\cdot n,\quad \textrm{on} \quad \partiala S_{0}, $$
the same conclusion as in Theorem \ref{thm:ws_s} holds.
\end{theorem}

\section{Proof of collision with slip conditions} \frak lambdabel{sec_slip}
This section is devoted to the proof of Theorem \ref{thm_slip}.  We assume $\beta_S$ and $\beta_{\Omega}$ are fixed non-negative slip lengths, and  $\rho_S > \rho_F$. Let $(S,u)$ be a weak solution over $(0,T)$,  as given by Theorem \ref{thm:ws_s}. We consider a solution up to its "maximal time of existence": that is $T=\infty$ if there is no collision,  or $T < \infty$ the time of collision.
We  assume that this weak solution satisfies A1-A4 for $t \in [0,T)$.   We denote by $h=h(t)$ the distance between the south pole of $S(t)$  and $x=0$.  By assumption A4,   when $h(t)$ is small enough,  it is equal to the distance between $S(t)$ and the boundary $\partialartial \Omega$. Our goal is to prove that $h(t)$ goes to zero in finite time. In other words, $T < \infty$, that is collision occurs in finite time.

To build up  our proof, we first need some general considerations on the connection between  the weak formulation \eqref{ws_s} and  the distance $h$,  for an arbitrary weak solution satisfying A4. We then design a special  test-function in order to prove  contact in finite time.

\subsection{Some features of the weak formulation \eqref{ws_s}}
Let $(S,u)$ a weak solution defined on $(0,T)$ and $\varphi \in \mathcal{T}_T^s$ of the form
\begin{equation} \frak lambdabel{eq_varphi}
\varphi  = \zeta(t) \varphi_{h(t)} \text{ with } \qquad  \varphi_S = \zeta(t)e_3\,.
\end{equation}
for some functions $\zeta \in \mathcal D([0,T))$ and $h \mapsto \varphi_{h}$ to be made precise.
Integrating by parts, we get that for any $q \in \mathcal D([0,T) \times \bar{\Omega}),$
for a.a. $t \in (0,T)$
\begin{eqnarray}o
&&2 \int_{F(t)} D(\varphi_F) : D(u_F)\\
&=& \int_{F(t)} \frak left( 2 D(\varphi_F) - q I \right) : \nablabla u_F\\
&=&-\int_{F(t)}(\Delta \varphi_F-\nablabla q)\cdot u_F+\int_{\partiala F(t)}(2D(\varphi_F)-qI)n\cdot u_F d\sigma\\
&=&-\int_{F(t)}(\Delta \varphi_F-\nablabla q)\cdot u_F+\int_{\partiala \Omega}(2D(\varphi_F)-qI)n\cdot u_F d\sigma+\int_{\partiala S(t)}(2D(\varphi_F)-qI)n\cdot u_F d\sigma\\
&=&-\int_{F(t)}(\Delta \varphi_F-\nablabla q)\cdot u_F+\int_{\partiala \Omega}(2 D(\varphi_F)-qI)n\cdot u_F d\sigma\\
&&\quad+\int_{\partiala S(t)}(2D(\varphi_F)-qI)n\cdot (u_F-u_S) d\sigma+\int_{\partiala S(t)}(2D(\varphi_F)-qI)n\cdot u_S d\sigma,
\end{eqnarray}o
Chosing also $q$ of the form $q = \zeta(t) q_{h(t)},$
the last term can be calculated by
\begin{eqnarray}o
\int_{\partiala S(t)}(2D(\varphi_F)-qI)n\cdot u_S d\sigma=h'(t) \zeta(t)\int_{\partiala S(t)}(2D(\varphi_{h(t)})-q_{h(t)}I)n\cdot e_3 d\sigma \triangleq \zeta(t) h'(t) n(h(t)).
\end{eqnarray}o
We emphasize that the domain $S(t)$ is completely fixed by the value of $h(t)$ (we shall denote $S_{h(t)}$ in what follows)
so that the last integral does depend on $h(t)$ only.
Combining all the equations with (\ref{ws_s}), we find
\begin{eqnarray}\frak lambdabel{drag}
&&\mu_F \int_0^T \zeta(t) h'(t)n(h(t)) \text{d$t$}\noindentnumber\\
&=&\int_0^T\int_{F(t)}\rho_F u_F\cdot\partiala_t\varphi_F +  \int_0^T\int_{S(t)}\rho_S u_S\cdot\partiala_t\varphi_S + \int_0^T\int_{F(t)}\rho_Fu_F\otimes u_F:\nablabla \varphi_F\noindentnumber\\
& -&\int_0^T\int_{F(t)}\rho_F ge_3\varphi_F -\int_0^T\int_{S(t)}\rho_S g e_3 \varphi_S + \int_{F_0}\rho_F u_{F,0}\varphi_F(0,\cdot) + \int_{S_0}\rho_S u_{S,0}\varphi_S(0,\cdot)\noindentnumber\\
&-&\f{\mu_F}{\beta_\Omega}\int_0^T\int_{P}(u_F\times n)\cdot(\varphi_F\times n)-\f{\mu_F}{\beta_S}\int_0^T\int_{\partiala S(t)}((u_F-u_S)\times n)\cdot((\varphi_F-\varphi_S)\times n)\noindentnumber\\
&+& \mu_F \Bigg( \int_0^T\int_{F(t)}(\Delta \varphi_F-\nablabla q)\cdot u_F - \int_0^T\int_{\partialartial \Omega}(2 D(\varphi_F)-qI)n\cdot u_F d\sigma\noindentnumber\\
&-&\int_0^T\int_{\partiala S(t)}(2 D(\varphi_F)-qI)n\cdot (u_F-u_S) d\sigma \Bigg)
\end{eqnarray}
This identity enables to describe the time evolution of $h$ through the function $n$ that we have to choose now.
To compute $n(h),$ we shall take advantage of the following identity:
\begin{eqnarray}
n(h)
&=&\int_{\partiala S_h}(2D (\varphi_{h})-q_{h}I)n\cdot \varphi_{h} d\sigma+\int_{\partiala S_h}(D(\varphi_{h})-q_{h}I)n\cdot (e_3-\varphi_{h}) d\sigma \noindenttag\\
&=& \int_{F_h}(\Delta \varphi_{h}-\nablabla q_{h}) \cdot \varphi_{h} dx+ 2 \int_{F_h}D(\varphi_{h}) : D(\varphi_{h}) dx \noindenttag \\
& - & \int_{\partiala \Omega}( 2D(\varphi_{h})-q_{h}I)n\cdot \varphi_{h} d\sigma+\int_{\partiala S_h}(D(\varphi_{h})-q_{h}I)n\cdot (e_3-\varphi_{h}) d\sigma. \frak lambdabel{n(h)}
\end{eqnarray}
where $F_h \triangleq \Omega \setminus \bar{S}_h.$ This identity holds true for all $h$ such that $S_h \subset\subset \Omega.$

\subsection{Construction of test-functions} \frak lambdabel{sec_tf}
We construct now the test function $\varphi_h$ and the pressure $q_h$ associated with
the solid particle $S_h$  frozen at distance $h$.  It will be defined for  $h \in (0, h_M)$, with $h_M \triangleq \sup_{t \in [0,T)} h(t)$.

When $h \to 0,$ a cusp arises in $F_h = \Omega \setminus \overline{S_h}.$ This cusp is contained in the domain :
$$
\Omega_{h,r} := \{ (x_1,x_2,x_3) \in F_h,  \: x_3 < \frac{1}{2} +h  \text{ and } |(x_1,x_2)| < r\}\,,
$$
for arbitrary $r \in (0,\frac{1}{2}).$ In particular, in all forthcoming estimates, we shall pay special attention to the region  $\Omega_{h,2 \delta}$, where $\delta$ was introduced in assumption A2.

To compute test-functions and associated pressures, we introduce cylindrical coordinates $(r,\thetaeta,z)$ associated with cartesian
coordinates $(x_1,x_2,x_3)$ in such a way that $x_3=z.$ The local basis reads $(e_r,e_{\thetaeta},e_z).$
First, we introduce an approximate solution of the steady Stokes equation in the domain $\Omega_{h,2\delta}$. It will satisfy locally the  slip boundary conditions \eqref{bc1}-\eqref{bc2},  with $u_S$ replaced by $e_z$:
$$\widetilde{\varphi}_h \\
=\f{1}{2}\frak left(-\partiala_z\Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right)x_1, -\partiala_z\Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right)x_2,\f{1}{r}\partiala_r\frak left(\Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right)r^2\right)\right)
$$
where
\begin{equation} \frak lambdabel{def_Phideb}
\gamma_s(r)=1-\sqrt{1-r^2},  \quad \Phi(r,t)=P_1(r)t+P_2(r)t^2+P_3t^3.
\end{equation}
Here
\begin{eqnarray}
P_1(r)&=&\f{6(2+\al_S)}{12+4(\al_S+\al_P)+\al_S\al_P},\\[4pt]
 P_2(r) &= & \f{3(2+\al_S)\al_P}{12+4(\al_S+\al_P)+\al_S\al_P},\\[4pt]
 P_3(r)&=&-\f{2(\al_S+\al_S\al_P+\al_P)}{12+4(\al_S+\al_P)+\al_S\al_P},
\end{eqnarray}
where
\begin{equation} \frak lambdabel{def_Phifin}
\al_P=\f{h+\gamma_s(r)}{\beta_\Omega},\quad \al_S= \frak left( \f{1}{\beta_S}  + 2\right)   (h+\gamma_s(r)).
\end{equation}
This approximation is inspired by the computations in \cite{DGVH12}.
We emphasize that the value of $\al_S$ does not match exactly the one in \cite{DGVH12}.
This modification is required so that further boundary estimates hold true (see \eqref{est:bdy}). {\em We stress that  notation $\partiala^k_z \Phi(r, \frac{z}{h+\gamma_s(r)})$, resp.  $\partiala^k_r \Phi(r, \frac{z}{h+\gamma_s(r)})$, stands for
$\displaystyle  \partiala_z^k\Psi(r,z)$, resp. Â  $\displaystyle \partiala_r^k\Psi(r,z), \quad   \Psi(r,z)  \triangleq     \Phi(r, \frac{z}{h+\gamma_s(r)}).$}
With this structure, $\tilde \varphi_h$ is divergence-free. Moreover, the function $\Phi$ is constructed so that
it satisfies the following boundary conditions:
\begin{itemize}
\item for $z=0$
\begin{eqnarray}
&&\Phi(r,0) = 0, \\
&& \partialartial_{zz} \Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right) - \dfrac{1}{\beta_{\Omega}} \partialartial_{z} \Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right) = 0
\end{eqnarray}
\item for $z= h + \gamma_S(r)$
\end{itemize}
\begin{eqnarray}
&&  \Phi(r,1) = 1\, \\
&& \partialartial_{zz} \Phi \frak left(r,\f{z}{h+\gamma_s(r)}\right)+\frak left( \dfrac{1}{\beta_{S}} + 2 \right)  \partialartial_{z} \Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right) = 0 \frak lambdabel{prop_Philast}
\end{eqnarray}
In the local cylindrical basis, the approximate solution $\widetilde{\varphi}_h$ reads:
\begin{equation} \frak lambdabel{tildephih}
\widetilde{\varphi}_h= \f{1}{2} \frak left[ -\partiala_z\Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right)r e_r +\f{1}{r}\partiala_r\frak left(\Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right)r^2\right) e_z  \right]\,.
\end{equation}
In particular, if we remark that $n$ satisfies $n = -r e_r + (1+h-z) e_z$ on $\partialartial S_h \cap \Omega_{h,1/2}$ and  $n = e_3$ on $\partialartial \Omega \cap \Omega_{h,1/2}$, we obtain (see the introduction of Appendix \ref{app} for further details) :
\begin{eqnarray}\frak lambdabel{bdy1}
\frak left\{
\begin{array}{rcl}
(\widetilde{\varphi}_h-e_3)\cdot n|_{\partiala S_h \, \cap \, \partialartial \Omega_{h,2\delta}}&=&0,\\[4pt]
\widetilde{\varphi}_h\cdot n|_{\partiala\Omega \, \cap \, \partialartial \Omega_{h,2\delta}}&=&0,\\[4pt]
\widetilde{\varphi}_h\times n|_{\partiala\Omega \, \cap \, \partialartial \Omega_{h,2\delta}}&=&-2\beta_{\Omega}(D(\widetilde{\varphi}_h)n)\times n|_{\partiala\Omega \, \cap \, \partialartial \Omega_{h,2\delta}}.
\end{array}
\right.
\end{eqnarray}
Thus, up to its extension  to the whole  of $\Omega$,  the function $\widetilde{\varphi}_h$ is a good candidate for applying the computations of the previous section. Such extension  $\varphi_h$ should satisfy $\varphi_h\vert_{S_h} = e_z$. It remains to define it into the whole of $F_h$.

To this end, we introduce a smooth  function $\partialhi=\partialhi(x)$ with compact support satisfying
\begin{eqnarray}o
\begin{array}{rcl}
\partialhi=1 && \mbox{ on  a $\frac{d_\delta}{2}$-neighborhood of the unit ball $B(0,1)$ },\\[4pt]
\partialhi = 0 &&   \mbox{ outside a $d_\delta$-neighborhood of $B(0,1)$}.
\end{array}
\end{eqnarray}o
We recall that $d_\delta$ was introduced in A4.
We also define a cut-off function $\chi : \mathbb R^3 \to [0,1]$ s.t.
\begin{equation} \frak lambdabel{eq_chi}
\chi=1  \textrm{ in } (-\delta,\delta)^3,\qquad \chi=0 \textrm{ outside }  (-2\delta,2\delta)^3.
\end{equation}
With these cut-off functions, we set (in cartesian coordinates):
\begin{eqnarray}\frak lambdabel{test_s}
&&
\varphi_h =\f{1}{2}\frak left(
\begin{array}{c}
  -x_1\partiala_z\frak left[(1-\chi(x))  \partialhi\frak left(x - (1+h) e_3\right) \, + \, \chi(x) \Phi(r,\f{z}{h+\gamma_s(r)})\right]\\[6pt]
   -x_2\partiala_z\frak left[(1-\chi(x))  \partialhi\frak left(x- (1+h) e_3\right) \, + \, \chi(x) \Phi(r,\f{z}{h+\gamma_s(r)})\right] \\[6pt]
  \f{1}{r}\partiala_r\frak left[r^2((1-\chi(x))  \partialhi\frak left(x-(1+h) e_3\right) \, + \, \chi(x) \Phi(r,\f{z}{h+\gamma_s(r)}))\right]
    \end{array}
\right) \textrm{ in } F_h,\noindentnumber\\
&&
\varphi_h=e_3 \textrm{ in } S_h.
\end{eqnarray}
By the definition of $\varphi_h$, we get
\begin{lemma} \frak lambdabel{lem_phih1}
The map $(h,x) \mapsto \varphi_h(x)$ is smooth on all domains:
$$
\{(h,x), h \in (0,h_M), x \in F_h\} \,,  \, \{(h,x) \in (0,h_M), x \in S_h\}\,,\,  \{(h,x), h \in [0,h_M), x \in F_h \setminus \Omega_{h,\delta}\}.
$$
Furthermore, there holds:
\begin{equation} \frak lambdabel{eq_contOmega}
\varphi_h\cdot n=0 \quad \text{ on $\partiala \Omega$}\\
\end{equation}
and
\begin{equation} \frak lambdabel{eq_contSh}
\varphi_h|_{F_h} \cdot n = e_3 \cdot n \quad \text{ on $\partiala S_h$}.
\end{equation}
\end{lemma}

This comes from direct calculations, that we leave to the reader for brevity.
We now state some refined estimates,  to be used  in the proof of {Theorem \ref{thm_slip}}. The whole point is to have an accurate bound on the blow-up of the test function as $h$ goes to zero.
\begin{lemma}\frak lambdabel{est:1}
There exist constants $0<c<C$ so that, when $h< h_M$ :
\begin{eqnarray}
\|\varphi_h\|_{L^2(F_h)} &\frak leq& C,\frak lambdabel{est:L2} \\
\|\nablabla \varphi_h\|^2_{L^2(F_h)}&\frak leq& C|\frak ln{h}|,\frak lambdabel{est:gradL2_1}\\
\|D(\varphi_h)\|^2_{L^2(F_h)}  &\geq&  c |\frak ln(h)|, \frak lambdabel{est:gradL2_2}
\end{eqnarray}
and
\begin{equation} \frak lambdabel{est:bdy}
\|2\beta_S (D({\varphi}_h) n)\times n+({\varphi}_h-e_3)\times n\|_{L^2(\partiala S_h)} \frak leq C,
\end{equation}
Furthermore, for $i=1,2,3,$ denoting by exponent $i$ cartesian coordinates, there holds:
\begin{eqnarray}
\|\int^{h+\gamma_s(r)}_0\partiala_h\varphi^i_hdz\|^2_{L^2(\partialartial\Omega \cap \partialartial \Omega_{h,\delta} )}&\frak leq& C|\frak ln{h}|, \frak lambdabel{dhpsi1}\\
 \|\int^{h+\gamma_s(r)}_z\partiala_h\varphi^i_hds\|_{L^2(\Omega_{h,\delta})}&\frak leq& C,\frak lambdabel{dhpsi2}\\
\|(h+\gamma_s(r))\partiala_i\int^{h+\gamma_s(r)}_z\partiala_h\varphi_h^i(x_1,x_2,s)ds\|_{L^2(\Omega_{h,\delta})}&\frak leq& C, \frak lambdabel{dhpsi3}
\end{eqnarray}
and for $(i,j) \in \{1,2,3\}^2$ and $x \in \Omega_{h,\delta}$:
\begin{eqnarray}
\frak left|\int^{h+\gamma_S(r)}_z(\partiala_i\varphi_h^j(x_1,x_2,s)+\partiala_j\varphi_h^i(x_1,x_2,s))ds\right|&\frak leq& C, \frak lambdabel{pctlP_zvarphi}\\
\frak left|(h+\gamma_S(r))\partiala_i\int^{h+\gamma_S(r)}_z(\partiala_i\varphi_h^j(x_1,x_2,s)+\partiala_j\varphi_h^i(x_1,x_2,s))ds\right|&\frak leq& C.
\end{eqnarray}
\end{lemma}
\noindentindent The proof of this lemma is postponed to Appendix \ref{app}.

When computing \eqref{drag}, we introduced a pressure $q.$ The aim was to smoothen
singularities in $\Delta \varphi_h$ which must arise when $h \to 0.$
The construction of this pressure is the content of the next lemma:
\begin{lemma}\frak lambdabel{est:2}
There exists a map $(h,x) \mapsto q_h$ which is smooth on the domains
$$
\{(h,x), h \in (0,h_M), x\in F_h  \} \text{ and } \{(h,x), h \in [0,h_M), x\in F_h  \setminus \Omega_{h,\delta} \}
$$
and which satisfies :
\begin{itemize}
\item there exist constants $0 < c < C < \infty$ such that :
\begin{eqnarray}o
 c|\frak ln{h}|\frak leq \int_{\partialartial S_h}(2D(\varphi_h)-q_hI)n\cdot (e_3-\varphi_h)d\sigma-\int_{\partialartial \Omega}(2D(\varphi_h)-q_hI)n\cdot \varphi_hd\sigma\frak leq C|\frak ln{h}| ;   \\
\end{eqnarray}o
\item there exists a positive constant $C$ such that, for any $h \in (0,h_M)$ and $v \in H^1(F_h)$
satisfying $v\cdot n =0$ on $\partialartial \Omega$, we have:
\begin{equation} \frak lambdabel{eq_deltaphi}
\frak left| \int_{F_h}(\Delta \varphi_h-\nablabla q_h)v\right|
\frak leq C \frak left( \|D(v)\|_{L^2(F_h)} +\|v\|_{L^2(\partiala \Omega)}\right).
\end{equation}
\end{itemize}
\end{lemma}
\noindentindent The proof of this lemma is also postponed to Appendix \ref{app}.

\subsection{Conclusion of the proof of Theorem \ref{thm_slip}} \frak lambdabel{sec_prfslip}

We are now in a position to prove Theorem \ref{thm_slip}. Combining the energy inequality \eqref{nrj_est_s} with the classical identity
$$ - \int_0^t \int_{F(t)} \rho_F g e_z \cdot u_F - \int_0^t \int_{S(t)} \rho_S g e_z \cdot u_S  = \dfrac{4(\rho_F - \rho_S)g\partiali}{3} (h(t)-h(0)) $$
we obtain easily  (remember that $\rho_S > \rho_F$):
\begin{multline}
\sup_{(0,T)} \int_{\Omega} |u(t)|^2  + \int_{0}^T \int_{F(t)} |D(u_F(t))|^2 \\
+\int_0^T\int_{\partialartial \Omega} |u_F \times n|^2 {\rm d}\sigma   +  \int_0^T \int_{\partialartial S(t)} |(u_F - u_S) \times n |^2 {\rm d}\sigma
\frak leq C_0. \frak lambdabel{eq_nrj} \end{multline}
This energy bound will be of constant use.

As mentioned before, we  take a special test function in the variational formulation \eqref{ws_s}. Namely, we take $\varphi(t,x) =  \zeta(t) \, \varphi_{h(t)}(x)$, with the field  $\varphi_h$ built in paragraph \ref{sec_tf}. From Lemma \ref{lem_phih1}, we have that $\varphi \in \mathcal T_T$ so that identity \eqref{drag} holds true.
The whole point is then to bound properly the r.h.s. of  (\ref{drag}):
\begin{eqnarray*}
RHS &\triangleq&\int_0^T\int_{F(t)}\rho_F u_F\cdot\partiala_t\varphi_F  +   \int_0^T\int_{S(t)}\rho_S u_S\cdot\partiala_t\varphi_S +\int_0^T\int_{F(t)}\rho_Fu_F\otimes u_F:\nablabla \varphi_F\noindentnumber\\
& -&\int_0^T\frak left(\int_{F(t)}\rho_F ge_z\varphi_F  + \int_{S(t)}\rho_S g e_z \varphi_S \right)+\frak left( \int_{F_0}\rho_F u_{F,0}\varphi_F(0,\cdot) + \int_{S_0}\rho_S u_{S,0}\varphi_S(0,\cdot)\right)\noindentnumber\\
&-&\mu_F  \int_0^T \int_{\partiala \Omega}(u_F\times n)\cdot  \frak left( \f{1}{\beta_\Omega}(\varphi_F\times n) +  (2 D(\varphi_F)-qI)n \right) d\sigma\\
&-&\mu_F \int_0^T  \int_{\partiala S(t)}((u_F-u_S)\times n) \cdot \frak left(  \f{1}{\beta_S}  (\varphi_F-\varphi_S)\times n)  + (2 D(\varphi_F)-qI)n \right) d\sigma   \noindentnumber\\
&+& \mu_F \int_0^T\int_{F(t)}(\Delta \varphi_F-\nablabla q)\cdot u_F \noindentnumber\\
&=&I_1 + I_2 + I_3 + I_4 + I_5 - \mu_F ( I_6 + I_7 + I_8),
\end{eqnarray*}
where we have applied the continuity of normal traces on $\partialartial S(t)$ and $\partialartial \Omega$ in the  boundary terms.
\begin{itemize}
\item
We first deal with
\end{itemize}
$$
I_3= \int_0^T\int_{F(t)}\rho_Fu_F\otimes u_F: \nablabla \varphi_F \triangleq  \int_0^T \rho_F\zeta(t)I_3(t).  \
$$
We introduce the functions $x \mapsto P_z \varphi^{i,j}_h(x)$ for $(i,j) \in \{1,2,3\}^2$ defined by:
\begin{eqnarray}o
 P_z \varphi^{i,j}_h(x)& =&  \int^{h+\gamma_S(r)}_{z} ( \partiala_i {\varphi}_h^j+\partiala_j{\varphi}_h^i)(x_1,x_2,s) ds   \,, \quad \forall\,  r < 1, \quad z < h + \gamma_s(r),\\
P_z \varphi^{i,j}_h(x) &=& \int^{h+1}_{z} ( \partiala_i {\varphi}_h^j+\partiala_j{\varphi}_h^i) (x_1,x_2,s) ds \quad \mbox{Â otherwise}.
\end{eqnarray}o
Let us stress that $P_z \varphi^{i,j}_h$ is smooth in $F_h$,  notably across $r=1$: it follows from the cancellation of  the symmetric gradient $\partiala_i {\varphi}_h^j +\partiala_j{\varphi}_h^i$ near  $(r=1,z=1+h)$ that both expressions coincide there.

Moreover,  by \eqref{pctlP_zvarphi}, there exists a constant $C$ independent of $t$ such that, for all $(i,j) \in \{1,2,3\}^2,$ there holds:
\begin{eqnarray}
|P_z \varphi_h^{i,j}(x_1,x_2,x_3)| +  |(h+\gamma_S(r)) \partialartial_{i} P_z \varphi_h^{i,j}(x_1,x_2,x_3)| \frak leq C, \quad \forall \, x \in \Omega_{h(t),\delta},  \frak lambdabel{Pzphi1}\\
|P_z \varphi_h^{i,j}(x_1,x_2,x_3)| +  | \partialartial_{i} P_z \varphi_h^{i,j}(x_1,x_2,x_3)| \frak leq C, \quad \forall \, x \in F(t) \setminus \Omega_{h(t),\delta}.\frak lambdabel{Pzphi2}
\end{eqnarray}
Then, by performing integration by parts, we obtain (summation on indices $i$ and $j$ is implicit):
\begin{eqnarray}o
I_3(t)&=&  - \int_{F(t)}u_F^iu_F^j \, \partiala_z[P_z \varphi^{i,j}_{h}] \: =  \: \int_{F(t)}\partiala_3(u_F^iu_F^j) P_z \varphi^{i,j}_{h} -\int_{\partiala \Omega} u_F^iu_F^j P_z \varphi^{i,j}_h n^3{\rm d}\sigma \\
&= & \int_{F(t)}((\partiala_3u_F^i+\partiala_iu_F^3)u_F^j+(\partiala_3u_F^j+\partiala_ju_F^3)u_F^i) P_z \varphi^{i,j}_h \\
& & \quad -  \int_{F(t)}(\partiala_i u_F^3 u_F^j+\partiala_j u_F^3 u_F^i) P_z\varphi^{i,j}_h  -\int_{\partiala \Omega} (u_F^iu_F^j)P_z \varphi^{i,j}_hn^3{\rm d}\sigma \\
&= &
\int_{F(t) }((\partiala_3u_F^i+\partiala_iu_F^3)u_F^j+(\partiala_3u_F^j+\partiala_ju_F^3)u_F^i) P_z \varphi^{i,j}_h\\
&&\quad + \int_{F(t)} \frak left( (\partiala_iu_F^j+\partiala_ju_F^i)u_F^3 P_z \varphi^{i,j}_h  + u_F^3u_F^j\partiala_i P_z \varphi^{i,j}_h   + u_F^3 u_F^i\partiala_j P_z \varphi^{i,j}_h \right) \\
&& \quad -\int_{\partiala \Omega} (u_F^iu_F^j n^3 + u_F^ju_F^3 n^i+ u_F^i u_F^3 n^j )P_z \varphi^{i,j}_h  {\rm d}\sigma.\\
\end{eqnarray}o
In this last identity, we apply \eqref{Pzphi1}-\eqref{Pzphi2} to obtain :
$$
\frak left|\int_{\partiala \Omega} (u_F^iu_F^j n^3 + u_F^ju_F^3 n^i+ u_F^i u_F^3 n^j )P_z \varphi^{i,j}_h  {\rm d}\sigma \right| \frak leq C   \|u_F(t)\|^2_{L^2(\partiala \Omega)}\,.
$$
Concerning the other terms, we have for instance :
\begin{eqnarray}o
\frak left| \int_{F(t) }u_F^3u_F^j\partiala_i P_z \varphi^{i,j}_h \right| &\frak leq&
\frak left| \int_{F(t) \cap \Omega_{h(t),\delta} }\dfrac{u_F^3}{h+\gamma_S(r)} u_F^j(h+\gamma_S(r))\partiala_i P_z \varphi^{i,j}_h \right|\\
&&\quad +
\frak left| \int_{F(t) \setminus \Omega_{h(t),\delta} } u_F^3u_F^j\partiala_i P_z \varphi^{i,j}_h \right|
\end{eqnarray}o

\begin{eqnarray}o
&\frak leq &
\|\f{u_F^3}{h+\gamma_S(r)}\|_{L^2(F(t) \cap \Omega_{h(t),\delta})} \|u_F\|_{L^2(F(t))} \|(h+\gamma_S(r))\partiala_i P_z \varphi^{i,j}_h\|_{L^{\infty}(F(t) \cap \Omega_{h(t),\delta}))}
\\
&&\quad  +
\|{u_F^3}\|_{L^2(F(t) \cap \{|(x_1,x_2)| < (1+d_\delta)\})} \|u_F\|_{L^2(F(t))} \|\partiala_i P_z \varphi^{i,j}_h\|_{L^{\infty}(F(t) \setminus \Omega_{h(t),\delta}))}.
\end{eqnarray}o
As $u^3_F(x) = 0$ for $x \in \{ |(x_1,x_2)| < 1+ d_\delta, x_3 =0\},$ a straightforward Poincar\'e inequality yields that
there exists a constant $C >0$ independent of time so that:
\begin{eqnarray}o
\|\f{u_F^3}{h+\gamma_S(r)}\|_{L^2(F(t) \cap \Omega_{h(t),\delta})} &\frak leq&  C \|\partiala_3 u_F^3\|_{L^2(F(t))} \\
\|{u_F^3}\|_{L^2(F(t) \cap \{ |(x_1,x_2)| < (1+d_\delta) \})} &\frak leq & C \frak left(   \|\partiala_3 u_F^3\|_{L^2(F(t))} +  \|u_F \times n \|_{L^2(\partiala \Omega)}\right).
\end{eqnarray}o
Introducing \eqref{Pzphi1}--\eqref{Pzphi2} then yields :
$$
|I_3(t)| \frak leq C \frak left[ \|D(u_F)\|^2_{L^2(F(t))} + \|D(u_F(t))\|_{L^2(F(t))} \|u_F(t)\|_{L^2(F(t))}  + \|u_F(t)\|^2_{L^2(\partiala \Omega)}\right],
$$
so that  \eqref{eq_nrj} implies there exists a constant $C$ depending only on initial data for which
$$
|I_3| \frak leq C(1+\sqrt{T}) \|\zeta\|_{L^{\infty}(0,T)}\,.
$$
\begin{itemize}
\item  To compute $I_1$, we split:
\end{itemize}
\begin{equation*}
I_1  =  \int_0^T\int_{F(t)}\rho_F u_F\cdot\partiala_t\varphi_F = \int_0^T\int_{F(t)}\rho_F u_F \zeta'(t) \varphi_{h(t)} + \int_0^{T} \int_{F(t)}\rho_F u_F \zeta(t) h'(t)\partiala_h\varphi_F \, \triangleq I_1^a + I_1^b.
\end{equation*}
Applying \eqref{est:L2} yields :
$$
|I_1^a| \frak leq C \|\zeta'\|_{L^1(0,T)} \|u\|_{L^{\infty}(0,T;L^2(\Omega))} \frak leq C_0  \|\zeta'\|_{L^1(0,T)}\,.
$$
Then, we have $I_1^b = \rho_F \int_0^t \zeta(t) h'(t) I_1^b(t)$ where, for all $t \in (0,T)$:
$$
I_1^b(t)= \sum_{i=1}^3 \int_{F(t)}u_F^i\cdot\partiala_h\varphi_{h(t)}^i.
$$
To bound this term, we define as previously $P_z \varphi_h^i$ which satisfies, for all $i=1,2,3$ :
\begin{eqnarray}o
 P_z \varphi^i_h(x)& =&  \int^{h+\gamma_S(r)}_{z} \partiala_h{\varphi}_h^i(x_1,x_2,s) ds \,, \quad \forall\,  r < 1, \quad z < h + \gamma_s(r),\\
P_z \varphi^i_h(x) &=& \int^{h+1}_{z}\partiala_h{\varphi}_h^i(x_1,x_2,s) ds  \quad \mbox{Â otherwise}.
\end{eqnarray}o
Again, $P_z \varphi^i_h$ is continuous in $F_h$, notably near $r=1$, as $\partiala_h \varphi_h$ vanishes near
$(r=1, z=1+h)$.

With \eqref{dhpsi2} together with \eqref{dhpsi3}, there is  a constant $C$ such that:
\begin{equation} \frak lambdabel{eq_Pzphi1}
\| P_z \varphi_h^{i} \|_{L^2(F(t))} +  \| (h+\gamma_S(r)) \partialartial_{i} P_z \varphi_h^{i}\|_{L^2(F(t)\cap \Omega_{h(t),2\delta})} + \|\partialartial_{i} P_z \varphi_h^{i}\|_{L^2(F(t)\setminus \Omega_{h(t),2\delta})}  \frak leq C,
\end{equation}
\begin{equation} \frak lambdabel{eq_Pzphi2}
\|P_z \varphi_h^i\|_{L^2(\partiala \Omega)} \frak leq C |\frak ln(h)|^{1/2}.
\end{equation}
From now on, we restrict implicitly  the integral $I_1^b(t)$ to $|(x_1,x_2)| < (1+ d_\delta)$ as $\partialartial_h \varphi_h$ vanishes elsewhere.
Performing integration by parts (summation over index $i$ is implicit), we obtain :
\begin{eqnarray}o
I_1^b(t)&=&- \int_{F(t)}  u_F^i \partiala_zP_z \varphi_h^i\\
& =  & \int_{F(t)} \partiala_3 u_F^i P_z \varphi_h^i -  \int_{\partiala \Omega} u_F^i P_z \varphi_h^i n^3  {\rm d}\sigma  \\
&\frak leq & \int_{F(t)} (\partiala_3 u_F^i + \partiala_i u_F^3 ) P_z \varphi_h^i  +   \int_{F(t)} u_F^3 \partialartial_ i P_z \varphi_h^i  - \int_{\partiala \Omega} (u_F^3 n^i + u_F^i n^3) P_z \varphi_h^i
\end{eqnarray}o
Relying on \eqref{eq_Pzphi1}-\eqref{eq_Pzphi2} and \eqref{eq_nrj}, we might then reproduce the arguments we used to compute $I_3,$ resulting in:
\begin{eqnarray}o
 |I_1^b| \frak leq  C \|\zeta\|_{L^{\infty}} \frak left( \int^T_0  |h'(t)| |\frak ln{h}|^{\frac{1}{2}}\|u_F \times n\|_{L^2(\partialartial \Omega)}+  \sqrt{T}\right).
\end{eqnarray}o
At this point, we remark that, given the computations of \cite[Section 3.2]{DGVH12},  $|\frak ln(h(t))|^{1/2}$ is the minimum dissipation energy
of functions $v \in H^1(F(t))$ satisfying $v \cdot n = e_3 \cdot n$ on $\partialartial S(t)$ and $v \cdot n = 0$ on $\partiala \Omega.$ Consequently, there exists a constant
$C$ independent of time such that:
$$
|\dot{h}| |\frak ln(h)|^{\frac{1}{2}} \frak leq C \frak left(  \int_{F(t)} |D(u_F(t))|^2 + \int_{\partialartial \Omega} |u_F \times n|^2 {\rm d}\sigma   +  \int_{\partialartial S(t)} |(u_F - u_S) \times n |^2 {\rm d}\sigma   \right)^{\frac{1}{2}},
$$
and \eqref{eq_nrj} implies:
$$
\int^T_0  |h'(t)| |\frak ln{h}|^{\frac{1}{2}}\|u_F \times n\|_{L^2(\partialartial \Omega)} \frak leq C\,.
$$
\begin{itemize}
\item  Next, we recall that
$\int_{F(t)}\rho_F ge_3\varphi_{h(t)} = - \dfrac{4 \partiali \rho_F g}{3}$,
which implies:
\end{itemize}
\begin{eqnarray}
I_4 = -\int^T_0\int_{F(t)}\rho_F ge_3\varphi_F-\int^T_0\int_{S(t)}\rho_S g e_3\varphi_S=  \dfrac{4 \partiali (\rho_F-\rho_S) g}{3} \int_0^T \zeta(t){\rm d}t.
\end{eqnarray}
\begin{itemize}
\item Then, we apply \eqref{bdy1} together with  \eqref{eq_nrj}. It leads to
\end{itemize}
\begin{eqnarray}o
|I_6|&=& \frak left|\int_0^T \zeta(t) \int_{\partiala \Omega}(u_F\times n)\cdot  \frak left( \f{1}{\beta_\Omega}(\varphi_h\times n) +  (2 D(\varphi_h)-qI)n \right) d\sigma\right|, \\
      &=&\frak left| \int_0^T \zeta(t) \int_{\partiala \Omega}(u_F\times n)\cdot  \frak left( \f{1}{\beta_\Omega}(\varphi_h\times n) +  2 D(\varphi_h) n \right) d\sigma  \right|,\\
      &\frak leq & C  \int_0^T |\zeta(t)| \|u_F\times n\|_{L^2(\partiala \Omega)} \frak leq C'  \|\zeta\|_{L^{\infty}(0,T)} \sqrt{T},
 \end{eqnarray}o
\begin{itemize}
\item By \eqref{est:bdy} together with \eqref{eq_nrj}:
\end{itemize}
\begin{eqnarray}o
|I_7| &=& \frak left| \int_0^T  \int_{\partiala S(t)}((u_F-u_S)\times n) \cdot \frak left(  \f{1}{\beta_S}  (\varphi_h-e_3)\times n)  + 2 D(\varphi_h) n \right) d\sigma\right| \\
       &\frak leq& C\int_0^T |\zeta(t)| \|(u_F-u_S)\times n\|_{L^2(\partiala S(t))} \|(\varphi_h - e_3)\times n) + 2 \beta_S D(\varphi_h) n\|_{L^2(\partiala S(t))}\text{d$t$}\\
  &\frak leq& C' \|\zeta\|_{L^{\infty}(0,T)} \sqrt{T}.
\end{eqnarray}o
\begin{itemize}
\item As for $I_8$, applying \eqref{eq_deltaphi}, we have that
\end{itemize}
\begin{eqnarray}o
I_8 = \int^T_0\int_{F(t)}(\Delta \varphi_F-\nablabla q)u_F&\frak leq& C\int^T_0 |\zeta(t)|\frak left( \|D(u_F)\|_{L^2(F(t))}+\|u_F\|_{L^2(\partiala \Omega)}\right)dt\\
					&\frak leq& C' \|\zeta\|_{L^{\infty}(0,T)} \sqrt{T}.
\end{eqnarray}o
\begin{itemize}
\item  Eventually, we get easily the following controls:
\end{itemize}
$$ | I_2 |Â  \: \frak le \:  C \, \| \zeta' \|_{L^1(0,T)}, \quad   I_5 \frak le C.$$

Combining all these estimates to bound $RHS$, we end up with:
\begin{multline} \frak lambdabel{finaldrag}
\mu_F \int_0^T  h'(t) \zeta(t) n(h(t))\text{d$t$} \\
= \dfrac{4 \partiali (\rho_F-\rho_S) g}{3} \int_0^T \zeta(t){\rm d}t + O (C_0 (\|\zeta\|_{L^{\infty}(0,T)}  + \|\zeta'\|_{L^1(0,T)}) (1+\sqrt{T}) )
\end{multline}

We conclude by introducing a sequence  $\zeta_n$ in ${\cal D}([0,T))$ such that
$$ \zeta_n \rightarrow 1_{[0,T)} \: \mbox{Â pointwise,}  \quad \mbox{with} \quad  \| \zeta_n \|_{L^\infty} \frak le C, \quad \| \zeta'_n \|_{L^1(0,T)} \frak le C. $$
A possible choice is $\zeta_n(t) = 1_{[0,T)}(t)Â - \chi(n(T-t))$, for some smooth $\chi$ compactly supported with $\chi = 1$ near $0$. We take $\zeta = \zeta_n$ in \eqref{finaldrag}, and let $n$ go to infinity, to get
\begin{equation} \frak lambdabel{eq_contradict}
\mu_F \int_{h(0)}^{h(T)} n(s)\text{d$s$} +  \dfrac{4 \partiali (\rho_S - \rho_F) g}{3} T  \frak leq  C_0 (1+\sqrt{T}).
\end{equation}
Introducing the bounds of Lemma \ref{est:1} and Lemma \ref{est:2} into \eqref{n(h)}, we show there exists a constant $C$ such that:
\begin{eqnarray}o
n(h)\frak leq C |\frak ln{h}| \quad \forall \, h < h_M,
\end{eqnarray}o
which implies that  there exists a constant $C$ for which:
\begin{eqnarray}o
\int^{h}_{h(0)} n(s) ds \geq -C\,, \quad \forall \, h < h_M\,.
\end{eqnarray}o
Hence, we obtain  a contradiction in \eqref{eq_contradict} if $T$ is large enough compared to $C$ and $C_0$ which
are fixed by $h_M$ and the initial data only.

\section{Proof of Theorem \ref{thm_mixed}} \frak lambdabel{sec_mix}
The proof is very close to the one performed above for slip boundary conditions.
Therefore, we only sketch the proof, insisting on  its few specific features.

\partialaragraph{Computation of the drag.}
Let $(S,u)$ be a weak solution given by Theorem \ref{thm:ws_m} on $[0,T)$. As before,  we consider a solution up to its "maximal time of existence": that is $T=\infty$ if there is no collision,  or $T < \infty$ the time of collision.  We assume that $(S,u)$  satisfies  A1-A4 over $[0,T)$. This time, we want to show that there is no collision, which means that $h$ does not go to zero in finite time (and thus $T=\infty$).  We shall consider a test function $\varphi \in \mathcal{T}_T^m$ of the form
\begin{equation} \frak lambdabel{eq_varphi_m}
\varphi  = \zeta(t) \varphi_{h(t)} \qquad \text{ with } \qquad  \varphi_S = \zeta(t)e_z\,.
\end{equation}
associated with a pressure $q \in \mathcal D([0,T) \times \bar{\Omega}),$ of the form
$$
q = \zeta(t) q_{h(t)}.
$$
The values of $\varphi_h$ and $q_h$ are given  below. As in the previous section,  we shall use functions $\zeta = \zeta_n$ approximating $1_{(0,T)}.$
Here and in what follows, we keep the convention that $\varphi_{F} = \varphi 1_{F(t)}$ and $\varphi_{S} = \varphi 1_{S(t)}\,.$
With  computations similar to those of the previous section, we obtain:
\begin{eqnarray}\frak lambdabel{drag-m}
&&\mu_F \int_0^T \zeta(t) h'(t)n(h(t)) \text{d$t$}\noindentnumber\\
&=&\int_0^T\int_{F(t)}\rho_F u_F\cdot\partiala_t\varphi_F  + \int_0^T\int_{F(t)}\rho_Fu_F\otimes u_F:\nablabla \varphi_F\noindentnumber\\
& -&\int_0^T\int_{F(t)}\rho_F ge_z\varphi_F -\int_0^T\int_{S(t)}\rho_S g e_z \varphi_S + \int_{F_0}\rho_F u_{F,0}\varphi_F(0,\cdot) + \int_{S_0}\rho_S u_{S,0}\varphi_S(0,\cdot)\noindentnumber\\
&-&\f{\mu_F}{\beta_\Omega}\int_0^T\int_{P}(u_F\times n)\cdot(\varphi_F\times n) \noindentnumber\\
&+& \mu_F \Bigg( \int_0^T\int_{F(t)}(\Delta \varphi_F-\nablabla q)\cdot u_F - \int_0^T\int_{\partialartial \Omega}(2 D(\varphi_F)-qI)n\cdot u_F d\sigma \Bigg)
\end{eqnarray}
Again, we compute $n(h)$ applying the identity:
\begin{eqnarray}
n(h)
&=& \int_{F_h}(\Delta \varphi_{h}-\nablabla q_{h}) \cdot \varphi_{h} dx+ 2 \int_{F_h}D(\varphi_{h}) : D(\varphi_{h}) dx \noindenttag \\
& - & \int_{\partiala \Omega}( 2D(\varphi_{h})-q_{h}I)n\cdot \varphi_{h} d\sigma. \frak lambdabel{n(h)-m}
\end{eqnarray}

\partialaragraph{Construction of the test-function.}
For the function $(x,h) \mapsto \varphi_h(x),$ we keep the structure of \eqref{test_s} with a test-function $\widetilde{\varphi}_h$
still of the form \eqref{tildephih}. The only point that differs is the choice of the coefficients in the  polynomial $\Phi.$ We set now:
$$
P_1(r) \triangleq \dfrac{6}{4+\alpha_P} \quad P_2(r) \triangleq  \dfrac{3\alpha_P}{4+ \alpha_P} \qquad P_3(r) \triangleq - 2 \dfrac{1+ \alpha_P}{4+ \alpha_P}\,,
$$
with $\alpha_P$ given by \eqref{def_Phifin}. We remark that this is equivalent to taking the limit $\alpha_S \to \infty$ in our choice of test-function in the previous
section.   With this choice, the function $\Phi$ satisfies boundary conditions:
\begin{itemize}
\item for $z=0$
\begin{eqnarray}
&&\Phi(r,0) = 0, \\
&& \partialartial_{zz} \Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right) - \dfrac{1}{\beta_{\Omega}} \partialartial_{z} \Phi\frak left(r,\f{z}{h+\gamma_s(r)}\right) = 0
\end{eqnarray}
\item for $z= h + \gamma_S(r)$
\end{itemize}
\begin{eqnarray}
&&  \Phi(r,1) = 1\, \\
&&\partialartial_z \Phi(r,1) = 0.
\end{eqnarray}
Similarly to the previous case, we provide also the following technical lemmas :
\begin{lemma} \frak lambdabel{lem_phih2}
The map $(h,x) \mapsto \varphi_h(x)$ is smooth on all domains :
$$
\{(h,x), h \in (0,h_M), x \in F_h\} \,,  \, \{(h,x) \in (0,h_M), x \in S_h\}\,,\,  \{(h,x), h \in [0,h_M), x \in F_h \setminus \Omega_{h,\delta}\}.
$$
Furthermore, there holds:
\begin{equation} \frak lambdabel{eq_contOmega_2}
\varphi_h\cdot n=0 \quad \text{ on $\partiala \Omega$}\\
\end{equation}
and
\begin{equation} \frak lambdabel{eq_contSh_2}
\varphi_h|_{F_h}= e_3  \quad \text{ on $\partiala S_h$}.
\end{equation}
\end{lemma}
\begin{lemma}\frak lambdabel{est_m:1}
There exist constants $0<c<C$ so that, when $h< h_M$ :
\begin{eqnarray}
\|\varphi_h\|_{L^2(F_h)} &\frak leq& C,\frak lambdabel{est_m:L2} \\
\|\nablabla \varphi_h\|^2_{L^2(F_h)}&\frak leq& \dfrac{C}{h},\frak lambdabel{est_m:gradL2_1}\\
\|D(\varphi_h)\|^2_{L^2(F_h)}  &\geq&  \dfrac{c}{h}, \frak lambdabel{est_m:gradL2_2}
\end{eqnarray}
Furthermore, for $i=1,2,3,$ denoting by exponent $i$ cartesian coordinates, there holds:
\begin{eqnarray}
\|\int^{h+\gamma_s(r)}_0\partiala_h\varphi^i_hdz\|_{L^2(\partialartial\Omega \cap \partialartial \Omega_{h,\delta} )}&\frak leq& C|\frak ln{h}|, \frak lambdabel{dhpsi1_2}\\
 \|\int^{h+\gamma_s(r)}_z\partiala_h\varphi^i_hds\|_{L^2(\Omega_{h,\delta})}&\frak leq& C,\frak lambdabel{dhpsi2_2}\\
\|(h+\gamma_s(r))\partiala_i\int^{h+\gamma_s(r)}_z\partiala_h\varphi_F^i(x_1,x_2,s)ds\|_{L^2(\Omega_{h,\delta})}&\frak leq& C, \frak lambdabel{dhpsi3_2}
\end{eqnarray}
and
\begin{itemize}
\item for $i \in \{1,2,3\}^2,$ and $x \in \Omega_{h,\delta}$:
\begin{eqnarray} \frak lambdabel{d3phiplus}
\Big| \partiala_i\varphi_h^3 +\partiala_3\varphi_h^i\Big|&\frak leq& \dfrac{Cr}{(h+ \gamma_S(r))^2}, \frak lambdabel{pctlP_zvarphi2}
\end{eqnarray}
\item for $(i,j) \in \{1,2\}^2,$ and $x \in \Omega_{h,\delta}$:
\begin{eqnarray}
\frak left|\int^{h+\gamma_S(r)}_z(\partiala_i\varphi_h^j(x_1,x_2,s)+\partiala_j\varphi_h^i(x_1,x_2,s))ds\right|&\frak leq& C, \frak lambdabel{pctlP_zvarphi2bis}\\
\frak left|(h+\gamma_S(r))\partiala_i\int^{h+\gamma_S(r)}_z(\partiala_i\varphi_h^j(x_1,x_2,s)+\partiala_j\varphi_h^i(x_1,x_2,s))ds\right|&\frak leq& C.
\end{eqnarray}
\end{itemize}
\end{lemma}
\begin{lemma}\frak lambdabel{est_m:2}
There exists a map $(h,x) \mapsto q_h$ which is smooth on the domain
$
\{(h,x), h \in (0,h_M), x\in F_h  \}
$
and which satisfies :
\begin{itemize}
\item there exists a constant positive constant $C$ such that :
\begin{eqnarray}o
\frak left|  \int_{\partialartial \Omega}(2 D(\varphi_h)-q_hI)n\cdot \varphi_hd\sigma \right| \: \frak leq \: C|\frak ln{h}|  ;   \\
\end{eqnarray}o
\item there exists a positive constant $C$ such that, for any $h \in (0,h_M)$ and $v \in H^1(\Omega)$
satisfying $v_{|_{S_h}} \in \mathcal R$ and $v\cdot n =0$ on $\partialartial \Omega$, we have:
\begin{equation} \frak lambdabel{eq_deltaphi2}
\frak left| \int_{F_h}(\Delta \varphi_h-\nablabla q_h)v\right|
\frak leq C \frak left( \|D(v)\|_{L^2(F_h)} +\|v\|_{L^2(\Omega)} \right).
\end{equation}
\end{itemize}
\end{lemma}
More details concerning these lemmas  are provided in Appendix  \ref{app_2}.

\partialaragraph{Conclusion.} Introducing the results of Lemma \ref{est_m:1} and Lemma \ref{est_m:2},
we compute again the right-hand side of \eqref{drag-m} with the help of energy estimates satisfied by $u.$ We only point out
two differences. First, there exists a universal constant $C$ so that
$$
 \|u_F\|_{L^2(\partialartial S(t))} = \| u_S \|_{L^2(\partiala S(t))}  \frak leq C \|u_S\|_{L^2(S(t))} \frak le C \|u\|_{L^2(\Omega)},
$$
by taking into account  that $u_S$  is a rigid vector field. Second, one term requires a new treatment (we drop summation over index $i \in \{1,2,3\} $ for simplicity):
\begin{eqnarray}o
& &\int_{F(t)}u_F^iu_F^3(\partiala_i\varphi_F^3+\partiala_3\varphi_F^i)\\
&=&\int_{F(t) \cap \Omega_{h(t),2\delta}} \chi \f{u^3_F}{\gamma_s(r)+h}u^i_F(\gamma_s(r)+h)(\partiala_i\varphi_F^3+\partiala_3\varphi_F^i) + O(\|u_F\|^2_{L^2})\\
&\frak leq&\|u^i_F(\gamma_s(r)+h)(\partiala_i\varphi_F^3+\partiala_3\varphi_F^i)\|_{L^2(F(t)\cap \Omega_{h(t),2\delta})}\|\f{u^3_F}{\gamma_s(r)+h}\|_{L^2(F(t) \cap \Omega_{h(t),2\delta})} + O(\|u_F\|^2_{L^2})\\
&\frak leq& C\|D(u_F)\|_{L^2(F(t))}\| \chi u^i_F(\gamma_s(r)+h)(\partiala_i\varphi_F^3+\partiala_3\varphi_F^i)\|_{L^2(F(t) \cap \Omega_{h(t),2\delta})} + O(\|u_F\|^2_{L^2(F(t))}).
\end{eqnarray}o
where $\chi$ is given by \eqref{eq_chi}. Because of \eqref{d3phiplus}, there holds
\begin{eqnarray}o
&&\|\chi u^i_F(\gamma_s(r)+h)(\partiala_i\varphi_F^3+\partiala_3\varphi_F^i)\|^2_{L^2(F(t) \cap \Omega_{h,2\delta})}\\
&\frak leq& \int_{F(t)}|u^i_F|^2\cdot \f{\chi}{\gamma_s(r)+h} \\
&=& \int_{F(t)} (\partiala_zu^i_F+\partiala_i u^3_F) \chi u^i_F \f{z-\gamma_s(r)-h}{\gamma_s(r)+h}-\int_{F(t)} \chi u^i_F \partiala_i u^3_F  \f{z-\gamma_s(r)-h}{\gamma_s(r)+h} \\
&& \quad + O(\|u_F\|^2_{L^2(F(t))}+  \|u_F\|^2_{L^2(\partiala \Omega)})\\
&=& \int_{F(t)}\dfrac{u^3_F}{\gamma_s(r)+h} u^i_F \chi (\gamma_s(r)+h) \partiala_i \frak left( \f{z-\gamma_s(r)-h}{\gamma_s(r)+h}\right)\\
 && \quad + O\Big(\|u_F\|^2_{L^2(F(t))}+  \|u_F\|^2_{L^2(\partiala \Omega)} + \|D(u_F)\|_{L^2(F(t))}\|u_F\|_{L^2(F(t))}\Big)\\
&=&O\Big(\|u_F\|^2_{L^2(F(t))}+  \|u_F\|^2_{L^2(\partiala \Omega)} + \|D(u_F)\|_{L^2(F(t))}\|u_F\|_{L^2(F(t))}\Big).
\end{eqnarray}o
which implies that
\begin{eqnarray}o
\frak left| \int_{F(t)}u_F^iu_F^3(\partiala_i\varphi_F^3+\partiala_3\varphi_F^i) \right|&=& O\Bigg( \|D(u_F)\|_{L^2(F(t))}(\|D(u_F)\| +\|u_F\|_{L^2(\partiala \Omega)})
											+ \|u_F\|^2_{L^2(F(t))}\Bigg)
\end{eqnarray}o
Combining with estimates similar to the previous section for the other terms, we arrive at:
\begin{equation} \frak lambdabel{eq_contradict2}
\frak left| \mu_F \int_{h(0)}^{h(T)} n(s)\text{d$s$}\right|  \frak leq  C_0 (1+T).
\end{equation}
Introducing the bounds of Lemma \ref{est_m:1} and Lemma \ref{est_m:2} into \eqref{n(h)}, we show there exists a constant $C$ such that:
\begin{eqnarray}o
n(h)\geq \dfrac{C}{h} \,, \quad \forall\, h \in (0, h_M),
\end{eqnarray}o
so that \eqref{eq_contradict2} yields finally :
\begin{eqnarray}o
|\frak ln(h(T))| \frak leq C_0(1+ T)\,, \quad \forall \, T >0.
\end{eqnarray}o
This ends the proof.

\partialaragraph{Acknowledgements.} 
The last author is supported by the Project Instabilit\'es hydrodynamiques funded partially by Mairie de Paris (program ÒEmergence(s)Ó)
and Fondation Sciences Math\'ematiques de Paris.

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\appendix

\section{Estimates on the test-function in the slip case}\frak lambdabel{app}
In this section, we collect estimates on the test-function of Section \ref{sec_slip}
as defined in \eqref{test_s}. For further computations, we define,
$$
\Psi(r,z) := \Phi\frak left(r, \dfrac{z}{h+\gamma_S(r)}\right)\,, \quad \forall  \, h \in (0,h_M),  \quad \forall \,(r,z) \in \Omega_{h,1}\,.
$$
In particular, we note that $\Psi(r,h+\gamma_S(r)) = 1$ so that:
\begin{eqnarray*}
\dfrac{\text{d}}{\text{d$r$}} \frak left[ \dfrac{r^2}{2}\Psi(r,h+\gamma_S(r))\right] &=& r \, \\
					&=& \partialartial_{r}\frak left( \dfrac{r^2}{2} \Psi\right)(r,h+\gamma_S(r))  + \dfrac{r}{\sqrt{1-r^2}} \dfrac{r^2}{2} \partialartial_z  \Psi (r,h+\gamma_S(r)).
\end{eqnarray*}
Thus, on $\partialartial S_h \cap \partialartial \Omega_{h,1},$ there holds :
\begin{eqnarray*}
n \cdot \tilde{\varphi}_h &=&  \dfrac{r^2}{2} \partialartial_z\Psi(r,h+\gamma_S(r)) + \dfrac{\sqrt{1-r^2}}{r}  \partialartial_{r}\frak left( \dfrac{r^2}{2} \Psi\right)(r,h+\gamma_S(r)),\\
&=&\sqrt{1-r^2} = e_3 \cdot n\,.
\end{eqnarray*}
Before going to estimates on $\varphi_h$, we first obtain a precise decription of the size of  $\Psi$ and its derivatives.
Here and in what follows we use  $a \frak lesssim b$ to denote that $a\frak leq Cb$ for a constant $C$ independant of relevant parameters.
\begin{proposition} \frak lambdabel{prop_Psi}
The following inequalities hold true, for all values of $h \in (0,h_M)$
and   $(r,z) \in \Omega_{h,\delta}\,:$
\begin{eqnarray}
|\Psi(r,z)| &\frak lesssim& 1 \,,  \\
|\partialartial_z \Psi(r,z)| &\frak lesssim&  \dfrac{1}{h + \gamma_S(r)}\,, \qquad |\partialartial_r \Psi(r,z)| \frak lesssim  \dfrac{r}{h+\gamma_S(r)} \,,\qquad \frak lambdabel{eq_dPsi1} \\
|\partialartial_{rr} \Psi(r,z)|& \frak lesssim& \dfrac{1}{h+\gamma_S(r)} \,, \qquad |\partialartial_{zz} \Psi(r,z)| \frak lesssim \dfrac{1}{h + \gamma_S(r)}\,, \qquad\\
|\partialartial_{rz} \Psi(r,z) + \dfrac{r}{h +\gamma_S(r)} \partialartial_z \Psi(r,z)| &\frak lesssim&  \dfrac{r}{h+\gamma_S(r)} \,, \frak lambdabel{drzpsi} \qquad \\
|\partialartial_{zzz} \Psi(r,z)| + |\partialartial_{rrz} \Psi(r,z)| & \frak lesssim&  \dfrac{1}{(h+\gamma_S(r))^2}\,, \\
|\partialartial_{rzz} \Psi(r,z)|  \frak lesssim \dfrac{r}{(h+\gamma_S(r))^2}, & &
|\partialartial_{rrr} \Psi(r,z)|  \frak lesssim  \dfrac{r}{(h+\gamma_S(r))^2} + \frac{1}{h+\gamma_s(r)}  \frak lambdabel{eq_dPsifin}
\end{eqnarray}
and also
\begin{eqnarray}
|\partialartial_{h} \Psi(r,z)| & \frak lesssim & \dfrac{1}{h+ \gamma_S(r)}\,, \\
|\partialartial_{rh} \Psi(r,z)|  & \frak lesssim & \dfrac{1}{(h+\gamma_S(r))} + \dfrac{r}{(h+\gamma_S(r))^2}\,,\\
|\partialartial_{zzh}\Psi(r,z)| + |\partialartial_{rrh} \Psi(r,z)| + |\partialartial_{hz} \Psi(r,z)| & \frak lesssim & \dfrac{1}{(h+\gamma_S(r))^2}\,,\\
|\partialartial_{rzh} \Psi(r,z)| & \frak lesssim &\dfrac{1}{(h+\gamma_S(r))^2} +   \dfrac{r}{(h+\gamma_S(r))^3}\,.
\end{eqnarray}
\end{proposition}
\begin{proof}
At first, we note that
\begin{eqnarray}o
|\partialartial_r \alpha_S| + |\partialartial_r \alpha_P| &\frak lesssim&r , \\
|\partialartial_{rrr} \alpha_S| + |\partialartial_{rr} \alpha_S | + |\partialartial_{rrr} \alpha_P| +  |\partialartial_{rr} \alpha_{P}| &\frak lesssim& 1,
\end{eqnarray}o
This yields, by \eqref{def_Phideb}-\eqref{def_Phifin}, that we have:
$$
 \qquad |\partialartial_r \Phi(r,t) | +|\partialartial_{rtt} \Phi(t,r)|  + |\partialartial_{tr} \Phi(t,r)| \frak lesssim r  \,,
 \quad \forall \, (r,t) \in (0,\delta) \times (0,1)\,.
$$
and
$$
|\partialartial_t \Phi(r,t)| + |\partialartial_{rrr}\Phi(t,r)|+ |\partialartial_{rrt} \Phi(t,r)|+  |\partialartial_{rr} \Phi(t,r)| \frak lesssim 1\,.
$$
We remark also that
$$
 |\partialartial_{ttt} \Phi(t,r)| + |\partialartial_{tt} \Phi(t,r)| \frak lesssim (\alpha_P+\alpha_S) \frak lesssim (h+\gamma_S(r)).
$$
We introduce these bounds implicitly in the following computations.

For instance, we have:
\begin{eqnarray*}
\partialartial_r \Psi(r,z)& =&  \partialartial_r \Phi\frak left(r , \dfrac{z}{h+\gamma_S(r)}\right) - \dfrac{z \gamma_S'(r)}{(h+\gamma_S(r))^2} \partialartial_t \Phi\frak left(r, \dfrac{z}{(h+\gamma_S(r)}\right)\\
			 & \frak lesssim &  \dfrac{r}{ (h+\gamma_S(r))}\,.
\end{eqnarray*}
We obtain similarly  \eqref{eq_dPsi1}--\eqref{eq_dPsifin} after tedious calculations.
The only computation we make precise is the one concerning $\partialartial_{rz} \Psi.$
Differentiating the last equality w.r.t. $z$, we get :
\begin{eqnarray*}
\partialartial_{rz} \Psi(r,z) &=& \dfrac{1}{h+\gamma_S(r)}\partialartial_{rt} \Phi  -  \dfrac{ \gamma_S'(r)}{(h+\gamma_S(r))^2} \partialartial_{t} \Phi
							-  \dfrac{z \gamma_S'(r)}{(h+\gamma_S(r))^3} \partialartial_{tt} \Phi,
\end{eqnarray*}
where :
$$
\partialartial_z \Psi(r,z) = \dfrac{1}{(h+\gamma_S(r))} \partialartial_{t} \Phi\,.
$$
Thus,
\begin{eqnarray*}
\partialartial_{rz} \Psi(r,z)  + \dfrac{r}{h +\gamma_S(r)} \partialartial_z \Psi(r,z) &=&
 \dfrac{1}{(h+\gamma_S(r))} \partialartial_{rt} \Phi
+ \dfrac{r  -  \gamma_S'(r)}{(h+\gamma_S(r))^2} \partialartial_{t} \Phi\\
&-& \dfrac{z \gamma_S'(r)}{(h+\gamma_S(r))^3} \partialartial_{tt} \Phi\,.
\end{eqnarray*}
where we dominate :
$$
\frak left| r-  \gamma_S'(r) \right| \frak lesssim r^3\,.
$$
Combining with the previous estimates on the derivative of $\Phi,$ we obtain the expected result.

Concerning derivatives w.r.t. $h$ the computations are similar. First, we note that :
$$
|\partialartial_{r}^k \partialartial_h \alpha_S(r)| + |\partialartial_{r}^k \partialartial_h \alpha_P(r)| \frak lesssim 1 \,, \quad \forall \, k \in \mathbb N\,.
$$
Correspondingly, we obtain that
$$
| \partialartial^k_{r} \partialartial^l_t \partialartial_h \Phi | \frak lesssim 1 \,, \quad \forall (k,l) \in \mathbb N^2\,.
$$
Then, we obtain, for instance :
\begin{eqnarray*}
\partialartial_{rh} \Psi(r,z)& =&  \partialartial_{rh} \Phi  - \dfrac{z}{(h+\gamma_S(r))^2}  \partialartial_{rt} \Phi
						+ 2 \dfrac{z \gamma_S'(r)}{(h+\gamma_S(r))^3} \partialartial_t \Phi\\
						&-& \dfrac{z \gamma_S'(r)}{(h+\gamma_S(r))^2} \partialartial_{th}  \Phi
						+ \dfrac{z^2 \gamma_S'(r)}{(h+\gamma_S(r))^4} \partialartial_{tt}  \Phi
						\end{eqnarray*}
in which, we plug previous estimates on $\Phi$ and its derivative. This yields :
$$
|\partialartial_{rh} \Psi(r,z)| \frak lesssim  \dfrac{1}{(h+\gamma_S(r))} +  \dfrac{r}{(h+\gamma_S(r))^2}\,.
$$
Other estimates are obtained similarly.
\end{proof}

The rest of  this appendix is devoted to the proofs of lemmas \ref{est:1} and \ref{est:2}.
They rely on the following lemma whose proof can be found in \cite{DGVH10}:
\begin{lemma}\frak lambdabel{lem:int}
Given $(p, q)\in (0,\infty)^2$, when $h$ goes to 0, the quantity
\begin{eqnarray}o
\int_0^\delta \f{|x|^pdx}{(h+|x|^{2})^q}
\end{eqnarray}o
behaves like\\
(1) $h^{\f{p+1}{2}-q}$, if $p+1< 2 q $,\\
(2) $\frak ln{h}$, if $p+1= 2 q,$\\
(3) $C$, if $p+1> 2 q$,\\
with $C$ a constant depending on $(p, q)$.
\end{lemma}

\subsection{Proof of Lemma \ref{est:1}}
First, we recall that, by construction $\varphi_h$ is smooth outside $\Omega_{h,\delta}.$ Thus we have get that
\begin{eqnarray}o
\|\nablabla^k \varphi_h\|_{L^\infty(F_h \setminus \Omega_{h,\delta})}\frak lesssim 1 \,, \quad \forall \, k \in \mathbb N\,.
\end{eqnarray}o
Hence, we focus on $\Omega_{h,\delta}$ and replace $\varphi_h$ with $\widetilde{\varphi}_h$ in what remains.
We keep notations of the previous lemma. In particular, we have by definition :
$$
\widetilde{\varphi}_h = \dfrac{1}{2} \frak left( -\partialartial_z \Psi(r,z) r e_r + \dfrac{1}{r}\partialartial_r\frak left[ r^2 \Psi(r,z)\right]e_{z}\right), \quad \text{ in $\Omega_{h,\delta}.$}
$$

\partialaragraph{\em Sobolev bounds.}
First, applying the bounds of the previous lemma, we get:
\begin{eqnarray}o
\|\widetilde{\varphi}_h\|_{L^2(\Omega_{h,\delta})} &\frak lesssim &\|r \partialartial_z \Psi\|_{L^2(\Omega_{h,\delta})} +  \|\dfrac{1}{r}\partialartial_r\frak left[ r^2 \Psi(r,z)\right]\|_{L^2(\Omega_{h,\delta})}\\
&\frak lesssim&  \frak left[  1 + \int^\delta_0\f{r^3{\rm d}r}{(r^2+h)}\right]^{\frac{1}{2}}\frak lesssim 1.
\end{eqnarray}o
Then, in cylindrical coordinates, we obtain:
\begin{equation} \frak lambdabel{est:nablaphipct}
|\nablabla \widetilde{\varphi}_h| \frak lesssim |\partialartial_z \Psi| + |r \partialartial_{zz} \Psi| + |\partialartial_r \Psi| +  |r\partialartial_{rr} \Psi| + \frac{r^2}{h+\gamma_s(r)},
\end{equation}
where we used \eqref{drzpsi} to get rid of $\partialartial_{rz} \Psi$ terms. Consequently :
$$
\|\nablabla \widetilde{\varphi}_h\|_{L^2(\Omega_{h,\delta})}^2 \frak lesssim  1+ \int_{0}^{\delta} \dfrac{r {\rm d}r}{(h+\gamma_S(r))}\, \frak lesssim |\frak ln(h)|\,.
$$
On the other hand,  we have
\begin{eqnarray}o
\|\partiala_2\varphi_h^2\|^2_{L^2(\Omega_{h,\delta})} &=& \dfrac{1}{4}\frak left\| \frak left(1 - \dfrac{r^2 |\sin(\thetaeta)|^2}{(h+ \gamma_S(r))^2}\right) \partialartial_z \Psi(r,z) \right\|^2_{L^2(\Omega_{h,\delta})}  \: +  \: O(1)
\end{eqnarray}o
Let now $\varepsilon >0$ small enough. There holds:
\begin{eqnarray}o
\|\partiala_2\varphi_h^2\|^2_{L^2(\Omega_{h,\delta})} &\geq & \frac{1}{8} \frak left\|  \partialartial_z \Psi(r,z) \right\|^2_{L^2(\Omega_{h,\delta}) \cap \{|\thetaeta| < \varepsilon\} }  \: +  \: O(1) \\
														& \geq & c \int_{0}^1 \int_{0}^{\delta} |\partialartial_t \Phi(r,t)|^2 \dfrac{r{\rm d}r \, {\rm d}t}{(h+\gamma_S(r))} \: +  \: O(1).
\end{eqnarray}o
where $c=c_\eps$ is a small positive constant. Finally, as $\partiala_t \Phi$ is bounded from below, we get
$$
\|\partiala_2\varphi_h^2\|^2_{L^2(\Omega_{h,\delta})}  \geq   c' \int_{0}^1 \int_{0}^{\delta}  \dfrac{r{\rm d}r \, {\rm d}t}{(h+\gamma_S(r))} + O(1)   \geq c |\frak ln h|.
$$
Combining the above estimate yields :
\begin{eqnarray}o
 c|\frak ln{h}|\frak leq \|D(\varphi_F)\|^2_{L^2(F(t))}\frak leq \|\nablabla \varphi_F\|^2_{L^2(F(t))}\frak leq C|\frak ln{h}|.
\end{eqnarray}o
This completes the proof of \eqref{est:L2}-\eqref{est:gradL2_2}.

\partialaragraph{\em Boundary estimates.} We proceed with estimate \eqref{est:bdy}. By standard identities of differential geometry (see \cite{BDG}), we have
\begin{eqnarray}o
2D(\widetilde{\varphi}_h)n\times n = {\partiala_n \widetilde{\varphi}_h} \times n+ (\widetilde{\varphi}_h-e_3) \times n.
\end{eqnarray}o
Thus, we only need to prove that
\begin{eqnarray}o
\|\beta_S {\partiala_n \widetilde{\varphi}_h} \times n + (\beta_S+1)\widetilde{\varphi}_h\times n \|_{L^2(\partiala S_h \, \cap \, \partialartial \Omega_{h,2\delta})}\frak leq C.
\end{eqnarray}o
In the aperture $\Omega_{h,2\delta}$ we recall that the normal to $\partialartial S_h$ directed toward $S_h$ satisfies
$n = -re_r + ((1+h)-z) e_z.$ This yields:
\begin{eqnarray*}
2\widetilde{\varphi}_h  \times n  	&=& \frak left( - \partialartial_z \Psi\ r e_r   + \dfrac{1}{r} \partialartial_r \frak left[ r^2 \Psi \right] e_z \right) \times ( -re_r + ((1+h)-z) e_z )\,,\\
			&=&  \frak left( r ((1+h)-z) \partialartial_z \Psi -      \partialartial_r \frak left[ r^2 \Psi \right]  \right) e_{\thetaeta}\,.
\end{eqnarray*}
Note here that, on $\partialartial S_h,$ we have $(1+h)-z - 1 \frak lesssim r^2$ and that
$$
\partialartial_r \frak left[ r^2 \Psi \right] \frak lesssim r = O(1)\,.
$$
Here and in the end of this proof $O(1)$ means bounded in $L^2(\partialartial S_h \, \cap \, \partialartial \Omega_{h,2\delta} )\,.$
This yields:
$$
\widetilde{\varphi}_h  \times n  = \dfrac{r}{2}    \partialartial_z \Psi e_{\thetaeta} + O(1) \,.
$$
On the other hand, in the aperture $\Omega_{h,\delta}$ we have:
\begin{eqnarray*}
2\, \partialartial_n \widetilde{\varphi}_h &=& \frak left( r \partialartial_{rz} (\Psi r)  - ((1+h)-z) r \partialartial_{zz}\Psi\right) e_r \\
				&+&\frak left(  -r \partialartial_r \frak left[\frac{1}{r} \partialartial_r[r^2\Psi]\right] +  ((1+h)-z)\partialartial_z \frak left[ \frac{1}{r} \partialartial_r [ r^2 \Psi ] \right] \right)e_z.
\end{eqnarray*}
Introducing the estimates of Proposition \ref{prop_Psi} we get in particular:

\begin{eqnarray*}
\partialartial_z \frak left[ \frac{1}{r} \partialartial_r [ r^2 \Psi ] \right] &=& 2 \partialartial_z \Psi + r \partialartial_{rz} \Psi \\
										&=& \frak left(2 - \dfrac{r^2}{h+\gamma_S(r)}  \right) \partialartial_z \Psi + O(1)=   \frac{2h}{h + \gamma_s(r)} \partialartial_z \Psi + O(1).
\end{eqnarray*}
Similarly, we have:
$$
r \partialartial_{rz} (\Psi r)  = r^2 \partialartial_{rz} \Psi + r \partialartial_z \Psi  = - r \partialartial_z \Psi + r \frak left( -\frac{r^2}{h + \gamma_s(r)}  + 2\right) \partiala_z \Psi  + O(1) = - r \partiala_z \Psi + O(1),
$$
using that
\begin{equation} \frak lambdabel{simplif}
 r \frak left( -\frac{r^2}{h + \gamma_s(r)}  + 2\right) \partiala_z \Psi = O\frak left( \frac{r h}{(h + \gamma_s(r))^2} \right)
 \end{equation}
is uniformly bounded in $L^2$ (use case 3 of Lemma \ref{lem:int} with $p=3$, $q=4$). Then,
$$
2 \partialartial_n \widetilde{\varphi}_h =  -( r \partialartial_z \Psi +  r \partialartial_{zz}\Psi) e_r \:   + \:   \frac{2h}{h+\gamma_s(r)} \partiala_z \Psi e_z \: + \:  O(1).
$$
Recalling that  $n = -re_r + ((1+h)-z) e_z$, and still applying \eqref{simplif}, we obtain:
$$
\partialartial_{n} \widetilde{\varphi}_h \times n = \dfrac{r}{2} \frak left(   \partialartial_{zz}\Psi + \partialartial_z \Psi\right) e_{\thetaeta}   + O(1).
$$
Finally, we obtain:
$$
\beta_S \partialartial_n \widetilde{\varphi}_h  \times n + (\beta_S + 1) \tilde{\varphi}_h \times n = \dfrac{r\beta_S}{2} \frak left(    \partialartial_{zz} \Psi  + \frak left(\dfrac{1}{\beta_S} + 2 \right) \partialartial_z \Psi\right) + O(1)\,.
$$
By our choice of $\alpha_S$, it follows that (see \eqref{prop_Philast})
\begin{equation*}
\partialartial_{zz} \Psi(r,h+\gamma_S(r))  + \frak left(\dfrac{1}{\beta_S} + 2 \right) \partialartial_z \Psi(r,h+\gamma_S(r))  = O(1).
 \end{equation*}
This yields \eqref{est:bdy}.

\partialaragraph{\em $h$-derivatives.} For $\partiala_h \widetilde{\varphi}_h$, we note that, for $i=1,2,3$ we have:
\begin{eqnarray}o
|\partiala_h \widetilde{\varphi}^i_h|\frak lesssim |r\partialartial_{zh} \Psi|  + |\partialartial_h \Psi| + |r \partialartial_{rh} \Psi| \frak lesssim \dfrac{r}{(h+\gamma_S(r))^2} + \dfrac{1}{(h+\gamma_S(r))}\,.
\end{eqnarray}o
Consequently, as integrating with respect to the $z$-variable reduces to remove one power of $(h+\gamma_S(r))$ in
the denominator, we get
$$
\|\int^{h+\gamma_s(r)}_z\partiala_h\widetilde{\varphi}^i_Fds\|^2_{L^2(F_h \cap \Omega_{h,\delta})} \frak lesssim 1+ \int_{0}^{\delta} \f{r^3 {\rm d}r}{(h + \gamma_S(r))} \frak lesssim 1\,.
$$
and
$$
\|\int^{h+\gamma_s(r)}_0\partiala_h\widetilde{\varphi}^i_Fdz\|^2_{L^2(\partialartial \Omega \cap \partialartial \Omega_{h,\delta}) }
\frak lesssim 1 +  \int^\delta_0\f{r^3{\rm d}r}{(h + \gamma_S(r))^2} \frak lesssim C|\frak ln{h}|\,.
$$
Finally, we also compute: for $i=1,2$:
\begin{eqnarray}o
|\partiala_i \partiala_h \widetilde{\varphi}_h| &\frak lesssim& |\partialartial_{zh} \Psi| + |r \partialartial_{rzh} \Psi| + |\partialartial_{rh} \Psi| + |r \partialartial_{rrh} \Psi| \\
							& \frak lesssim & \dfrac{1}{(h+\gamma_S(r))^2}
\end{eqnarray}o
so that for the same reason as above, we have:
\begin{eqnarray}o
|(h+\gamma_s(r))\partiala_i\int^{h+\gamma_s(r)}_z\partiala_h\widetilde{\varphi}_h^i(x_1,x_2,s)ds| \frak lesssim 1 \,, \quad \text{ for $i=1,2$}\, \\
\end{eqnarray}o
and
$$
\|(h+\gamma_s(r))\partiala_i\int^{h+\gamma_s(r)}_z\partiala_h\widetilde{\varphi}_F^i(x_1,x_2,s)ds\|_{L^2(F_h\cap \Omega_{h,\delta})}\frak lesssim 1,\quad \text{ for $i=1,2,3$}\, .
$$
This ends the proof of \eqref{dhpsi1_2}--\eqref{dhpsi3_2}.

\partialaragraph{\em Pointwise bounds.}
We easily get that on $\Omega_{h,\delta}$
\begin{eqnarray}o
|\nablabla \varphi_h| \frak lesssim |r\partialartial_{rz}\Psi| + |\partialartial_z \Psi| + |r\partialartial_{rr} \Psi| + |\partialartial_{r}\Psi| + |r\partialartial_{zz}\Psi| \frak lesssim \dfrac{1}{h + \gamma_S(r)},
\end{eqnarray}o
from which we conclude:
$$
\frak left|\int^{h+\gamma_S(r)}_z(\partiala_i\varphi_h^j(x_1,x_2,s)+\partiala_j\varphi_h^i(x_1,x_2,s))ds\right|\frak lesssim 1, \text{ for $(i,j) \in \{1,2,3\}^2.$}
$$
Similarly, we get:
\begin{eqnarray}o
|\partialartial_{1} \nablabla \varphi_h| + |\partialartial_2 \nablabla \varphi_h| \frak lesssim |r \partialartial_{rrr}\Psi| + |\partialartial_{rr} \Psi| + |r \partialartial_{rrz} \Psi| + |\partialartial_{rz} \Psi| + |  \partiala_{zz} \Psi | \frak lesssim \, \dfrac{r}{(h+\gamma_S(r))^2} \: + \: \frac{1}{h+\gamma_s}.
 \end{eqnarray}o
and we conclude:
$$
\frak left|(h+\gamma_S(r))\partiala_i\int^{h+\gamma_S(r)}_z(\partiala_i\varphi_F^j(x_1,x_2,s)+\partiala_j\varphi_F^i(x_1,x_2,s))ds\right| \frak lesssim 1.
$$
This ends the proof of our lemma.

\subsection{Proof of Lemma \ref{est:2}}
\partialaragraph{\em Boundary  integrals} We focus on the domain $\Omega_{h,\delta}$ and we replace $\varphi_h$ with $\widetilde{\varphi}_h$
in the proof because everything is bounded elsewhere. First, it turns out that the first identity is independant of $q_h.$
Indeed, applying boundary conditions
\eqref{bdy1} we obtain :
\begin{eqnarray}o
\int_{\partialartial \Omega_{h,\delta} \cap \partiala \Omega } (2 D(\widetilde{\varphi}_h) - q_h) n \cdot \widetilde{\varphi}_h \text{d$\sigma$}
&=&
\int_{\partialartial \Omega_{h,\delta} \cap \partiala \Omega } [(2 D(\widetilde{\varphi}_h) n) \times n]   \cdot ( \widetilde{\varphi}_h \times n) \text{d$\sigma$}\\
&=&
-\dfrac{1}{\beta_{\Omega}} \int_{\partiala \Omega_{h,\delta} \cap \partiala \Omega} |\widetilde{\varphi}_h \times n|^2 \text{d$\sigma$}\,.
\end{eqnarray}o
Similarly, introducing boundary conditions \eqref{bdy1} and estimate \eqref{est:bdy}, we compute:
\begin{multline*}
\int_{\partialartial \Omega_{h,\delta} \cap \partialartial S_h } (2 D(\widetilde{\varphi}_h) - q_h) n \cdot (e_3 - \widetilde{\varphi}_h) \text{d$\sigma$}
=
\int_{\partialartial \Omega_{h,\delta} \cap \partialartial S_h } [(2 D(\widetilde{\varphi}_h) n) \times n]   \cdot ( (e_3 -  \widetilde{\varphi}_h) \times n) \text{d$\sigma$}\\
=
\dfrac{1}{\beta_{S}} \int_{\partiala \Omega_{h,\delta} \cap \partialartial S_h} |( e_3 - \widetilde{\varphi}_h) \times n|^2 \text{d$\sigma$} + O(\|( e_3 - \widetilde{\varphi}_h) \times n\|_{L^2(\partialartial S_h \cap \Omega_{h,\delta})})\,.
\end{multline*}
On the boundaries, there holds
\begin{eqnarray}o
\widetilde{\varphi}_h =  - \dfrac{P_1(r)}{(h+\gamma_S(r))}re_r \quad \textrm{in} \quad \partiala \Omega_{h,\delta}\cap \partiala \Omega,
\quad
|\widetilde{\varphi}_h|  \frak lesssim \dfrac{r}{(h+\gamma_S(r))}   \quad \textrm{in} \quad \partiala \Omega_{h,\delta}\cap \partiala S_h,
\end{eqnarray}o
which implies that
\begin{eqnarray}o
|\frak ln(h)| \frak lesssim \int^\delta_0 \f{r^3{dr}}{(h+\gamma_S(r))^2} \frak lesssim \|\widetilde{\varphi}_h \times n \|^2_{L^2(\partiala \Omega_{h,\delta}\cap \partiala \Omega)} =  \|\widetilde{\varphi}_h \|^2_{L^2(\partiala \Omega_{h,\delta}\cap \partiala \Omega)}  \end{eqnarray}o
and
which implies that
\begin{eqnarray}o
|\frak ln{h}|\frak lesssim \int_{\partialartial \Omega_{h,\delta} \cap \partialartial S_h } (2 D(\widetilde{\varphi}_h) - q_h) n \cdot (e_3 - \widetilde{\varphi}_h) \text{d$\sigma$}
  - \int_{\partiala \Omega \cap \partiala\Omega_{h,\delta}}(D(\widetilde{\varphi}_h)-q_hI)n\cdot \varphi_h d\sigma
 \frak lesssim |\frak ln(h)|.
 \end{eqnarray}o

\partialaragraph{\em Construction of $q_h$.}
Again, we focus on the domain $\Omega_{h,\delta}$  where $\varphi_h=\widetilde{\varphi}_h$ to compute $q_h.$
The construction is extended to the whole $F_h$ by a standard truncation argument.

Given $h>0,$ let define
\begin{equation}\frak lambdabel{q}
2q_h \triangleq - \frak left( r \partialartial_{rz} \Psi + 2 \partialartial_z \Psi + \int^r_0\partiala_{zzz} \Psi(s,z)sds \right). \noindentnumber\\
\end{equation}
Obviously, the map $(h,x) \mapsto q_h(x)$ is $C^1$ on $\{ h \in (0,h_M), \; x \in F_h\}.$
and we have $\Delta \widetilde{\varphi}_h - \nablabla q_h = f_h e_3$ where
$$
|f_h| \frak lesssim |r\partialartial_{rrr} \Psi| + |\partialartial_{rr} \Psi| + \frak left|\dfrac{\partialartial_r \Psi}{r} \right| + |r \partialartial_{rzz}\Psi| + |\partialartial_{zz} \Psi| \frak lesssim \dfrac{1}{(h+\gamma_s(r))}
$$
Then, for arbitrary $h \in (0,h_M)$ and $v \in H^1(F_h)$ we introduce $\chi$ the truncation function as defined in  \eqref{eq_chi} and
 we split :
 $$
  \int_{F_h}(\Delta \varphi_h-\nablabla q_h)v  = \int_{F_h} \chi (\Delta \varphi_h-\nablabla q_h)v + \int_{F_h} (1-\chi) (\Delta \varphi_h-\nablabla q_h)v  = I + II.
 $$
Then, we have:
\begin{eqnarray}o
\frak left| I  \right| &=& \int_{F_h \cap \Omega_{h,2\delta}} |\chi f_h|| v_3| \\
&\frak leq & \| (h+\gamma_S) f_h\|_{L^2(F_h \cap \Omega_{h,2\delta})} \| \f{v_3}{\gamma_s(r)+h} \|_{L^2(F_h \cap \Omega_{h,2\delta})}
\end{eqnarray}o
where a standard Poincar\'e inequality yields:
\begin{eqnarray}o
\|\f{v_3}{\gamma_s(r)+h}\|_{L^2(F_h \cap \Omega_{h,2\delta})}\frak lesssim \|\partiala_3v_3\|_{L^2(F_h \cap \Omega_{h,\delta})} \frak lesssim  \|D(v)\|_{L^2(F_h \cap \Omega_{h,\delta})}.
\end{eqnarray}o
As for the second integral, we introduce $P_z R^i$ for $i=1,2,3,$ which satisfy :
\begin{eqnarray}
P_z R^i &=& \int_z^{h+\gamma_s(r)} (1-\chi) (\Delta \varphi^i_h-\partiala_i q_h) \quad \text{for $r < 1$, $\: z < r + \gamma_s(r)$}\,\\
P_z R^i &=& \int_z^{h+1}(1-\chi) (\Delta \varphi^i_h-\partiala_i q_h)   \quad \text{ otherwise}.
\end{eqnarray}
Repeating the computations of $I_1(t)$ and $I_2(t)$ in Section \ref{sec_prfslip}, we obtain (the summation over index
$i$ is implicit) :
\begin{eqnarray}o
| II | =  \int_{F_h} (\partialartial_3 v^i + \partialartial_i v^3 ) P_z R^i + \int_{F_h}  v^3 \partialartial_i P_z R^i - \int_{\partiala \Omega} (v^3 n^i + v^in^3) P_z R^i \text{d$\sigma$}\,.
\end{eqnarray}o
As $(\Delta \varphi_h-\nablabla  q_h)$ is uniformly bounded outside $\Omega_{h,\delta},$
applying a Cauchy Schwarz inequality to dominate $K_2$ and combining with $K_1$ concludes the proof :
$$
\frak left| \int_{F_h}(\Delta \varphi_h-\nablabla q_h) \cdot v \right|  \frak lesssim  \|D(v)\|_{L^2(F_h)} + \|v\|_{L^2(\partiala \Omega)}.
$$

\section{Estimates on the test function in the mixed case}\frak lambdabel{app_2}
In this appendix, we collect estimates on the test function of Section \ref{sec_mix}.
As previously, we define,
$$
\Psi(r,z) := \Phi\frak left(r, \dfrac{z}{h+\gamma_S(r)}\right)\,, \quad \forall  \, h \in (0,h_M),  \quad \forall \,(r,z) \in \Omega_{h,1}\,.
$$
The only point which differs from the test-function in the slip case is the decription of this $\Psi.$
Combining these estimates in the same manner as in the previous appendix, we obtain Lemma \ref{est_m:1}.
As the computations are completely similar, all the proofs are left to the reader.
\begin{proposition} \frak lambdabel{prop_Psi_mix}
The following inequalities hold true, for all  $h \in (0,h_M)$
and   $(r,z) \in \Omega_{h,\delta}\,:$
\begin{eqnarray}
|\Psi(r,z)| &\frak lesssim& 1 \,,  \\
|\partialartial_z \Psi(r,z)| &\frak lesssim&  \dfrac{1}{h + \gamma_S(r)}\,, \qquad |\partialartial_r \Psi(r,z)| \frak lesssim  \dfrac{r}{h+\gamma_S(r)} \,,\qquad  \\
|\partialartial_{rr} \Psi(r,z)|& \frak lesssim& \dfrac{1}{h+\gamma_S(r)} \,, \qquad |\partialartial_{zz} \Psi(r,z)| \frak lesssim \dfrac{1}{(h + \gamma_S(r)^2}\,, \qquad\\
|\partialartial_{rz} \Psi(r,z)| &\frak lesssim&  \dfrac{r}{(h+\gamma_S(r))^2} \,, \qquad |\partialartial_{zzz} \Psi(r,z)|  \frak lesssim  \dfrac{1}{(h+\gamma_S(r))^3}  \\
|\partialartial_{rrz} \Psi(r,z)| & \frak lesssim&  \dfrac{1}{(h+\gamma_S(r))^2}\,, \qquad |\partialartial_{rzz} \Psi(r,z)|  \frak lesssim \dfrac{r}{(h+\gamma_S(r))^3}\,, \\
|\partialartial_{rrr} \Psi(r,z)| & \frak lesssim&  \dfrac{r}{(h+\gamma_S(r))^2}\,,\qquad |\partialartial_{rrrz} \Psi(r,z)|  \frak lesssim \dfrac{r}{(h+\gamma_S(r))^3}\,, \\
\end{eqnarray}
and also
\begin{eqnarray}
|\partialartial_{h} \Psi(r,z)| & \frak lesssim & \dfrac{1}{h+ \gamma_S(r)}\,, \qquad
|\partialartial_{rh} \Psi(r,z)|   \frak lesssim  \dfrac{r}{(h+\gamma_S(r))^2}\,,\\
|\partialartial_{rrh} \Psi(r,z)|  & \frak lesssim & \dfrac{1}{(h+\gamma_S(r))^2}\,,
\qquad  |\partialartial_{hz} \Psi(r,z)|  \frak lesssim  \dfrac{1}{(h+\gamma_S(r))^2} \\
|\partialartial_{rzh} \Psi(r,z)| & \frak lesssim &  \dfrac{r}{(h+\gamma_S(r))^3} + \dfrac{1}{(h+\gamma_S(r))^2} \\
|\partialartial_{zzh}\Psi(r,z)| &\frak leq& \dfrac{1}{(h+\gamma_S(r))^3} \,.
\end{eqnarray}
\end{proposition}

We also need a different way to construct the pressure $q_h$ in order to prove Lemma \ref{est_m:2}. Again, we construct $q_h$
only in the aperture domain $\Omega_{h,\delta},$ the extension to $F_h$ being done by a standard
truncation. So, for $h>0$ we set :
$$
2 q_h \triangleq  r \partialartial_{rz} \Psi + 2 \partialartial_z \Psi - \int_0^r \partialartial_{zzz} \Psi s \text{d$s$}\,.
$$
In this way, we obtain $\Delta {\varphi}_h - \nablabla q_h =   f^r e_r +  f^z e_z$ where the $f_i$'s are bounded outside
$\Omega_{h,\delta}$ and, in $\Omega_{h,\delta},$ we have:
\begin{eqnarray}
|f^z| &\frak lesssim& |r \partialartial_{rrr} \Psi| + |\partialartial_{rr} \Psi| + \frak left|  \dfrac{\partialartial_r \Psi}{r} \right| \frak lesssim \dfrac{1}{(h+\gamma_S(r))} \frak lambdabel{f3}\\
|f^r| &\frak lesssim & |r \partialartial_{rrz} \Psi| + |\partialartial_{rz} \Psi| \frak lesssim \dfrac{r}{(h+\gamma_S(r))^2}				\frak lambdabel{fp1}\\
|\partialartial_r f^r| + \frak left| \dfrac{f^r}{r} \right|&\frak lesssim&  |r\partialartial_{rrrz} \Psi| +  |\partialartial_{rrz} \Psi| + \frak left| \dfrac{\partialartial_{rz}\Psi}{r}\right| \frak lesssim \dfrac{1}{(h+\gamma_S(r))^2}. \frak lambdabel{fp2}
\end{eqnarray}
Then, for any $v \in H^1(\Omega)$ such that $v \cdot n = 0$ on $\partiala \Omega$ and $v \in \mathcal R$ on $S_h$ we have :
$$
\int_{F_h} (\Delta \varphi_h - \nablabla q_h) \cdot v = \int_{F_h} f^z v^3 + \int_{F_h}  f^r e_r \cdot v .
$$
First, we apply  Poincar\'e inequality, as $v^3 = 0$ on $\partiala \Omega \cap \partiala \Omega_{h,\delta},$ together with \eqref{f3}. This yields:
\begin{eqnarray}o
\frak left| \int_{F_h} f^z v_3 \right| &\frak lesssim&  \frak left| \int_{F_h \cap \Omega_{h,\delta}} f^z (h+\gamma_S(r)) \dfrac{v_3}{(h+\gamma_S(r))}\right| +\|v\|_{L^2(F_h)}  \\
						&\frak lesssim & \|D(v)\|_{L^2(F_h)} + \|v\|_{L^2(F_h)}
\end{eqnarray}o
For the second integral:  we write $f^r e_r  = f^1 e_1 + f^2 e_2$, and split the integral, with $\chi$ as defined in \eqref{eq_chi} :
$$
\int_{F_h}  f^i v_i =  \int_{F_h} \chi  f^i   v^i + O(\|v\|_{L^2(F_h)})\,,
$$
as  $f^r$ is uniformly bounded outside $\Omega_{h,\delta}.$
Then, we introduce  $P_z f^i,$ for $i =1,2,$ which satisfies:
$$
P_z f^i  = \int_0^z \chi f^i
$$
Through integration by parts, we get (with the Einstein convention of repeated indices):
\begin{eqnarray}o
 \int_{F_h} \chi f^i v^i &=& \int_{F_h \cap \Omega_{h,2\delta}} P_z f^i (\partialartial_3 v^i + \partialartial_i v^3) + \int_{F_h \cap \Omega_{h,2\delta}}  \dfrac{v_3}{h+\gamma_S(r)} (h+\gamma_S(r)) \frak left( \partiala_i P_z f^i  \right) \, \\
 				 	&& - \int_{\partialartial S_h \cap \Omega_{h,2\delta}} P_z f^i  (v^i n^3 + v^3 n^i) \: \triangleq \: I + II + III
\end{eqnarray}o
Thanks to the bounds \eqref{fp1}-\eqref{fp2}, we get:
$$ |I| + |II| \: \frak lesssim \: \|Â D(v) \|_{L^2(F_h)}. $$
As regards the last integral, we simply write, because $v_{|_{S_h}} \in {\cal R},$
$$ | III | \: \frak lesssim \: \| P_z f^i \|_{L^1(\partiala S_h)} \, \| v \|_{L^\infty(\partiala S_h)}  \: \frak lesssim \:  \|Â v \|_{L^2(S_h)} \frak lesssim  \|Â v \|_{L^2(\Omega)} $$
This ends the proof.

\end{document}