text:
\section{Introduction}\label{sec:intro}

The geodynamo is the process by which the Earth's magnetic field is created and sustained by the motion of the fluid core which is composed of a rapidly rotating, density stratified, electrically conducting fluid. Recently Moffat \cite{Moffatt08} proposed a model for the geodynamo which is a reduction of the full magnetohydrodynamic system of PDE. This model can be viewed as a nonlinear active scalar equation in three dimensions. An explicit operator $\boldsymbol M[\Theta]$ encodes the physics of the geodynamo and produces the divergence free drift velocity $\boldsymbol U(t,\boldsymbol x)$ from the scalar ``buoyancy'' field $\Theta(t,\boldsymbol x)$. We call this active scalar equation the magnetogeostrophic equation $(MG)$. It has some features in common with the much studied two dimensional surface quasigeostrophic equation $(SQG)$ (see \cite{CaffarelliVasseur10,ChaeConstantinCordobaGancedoWu11,ChaeConstantinWu11,ConstantinWu99,CordobaCordoba04,Ju04,KiselevNazarovVolberg07, Rusin11, Wu04} and references therein). However the $(MG)$ equation has a number of novel and distinctive features due to the strongly singular and anisotropic nature of the operator $\boldsymbol M$.

The {\em critically diffusive} $(MG)$ equation is defined by the following system
\begin{align}
  &\partial_t \Theta + \boldsymbol U \cdot \nabla \Theta  =S + \kappa \Delta \Theta \label{eq:1}\\
  &\nabla \cdot \boldsymbol U = 0\label{eq:2}
\end{align}
where $\kappa \geq 0$ is a physical parameter, and $S(t,\boldsymbol x)$ is a $C^\infty$-smooth, bounded in time source term. The velocity  $\boldsymbol U$ is divergence-free, and is obtained from $\Theta$ via
\begin{align}
U_j =  M_j  \Theta \label{eq:3},
\end{align}
for all $j \in \{1,2,3\}$, where the $M_j$ are Fourier multiplier operators with symbols given explicitly by
\begin{align}
& \hat{M}_1(\boldsymbol k) = \frac{2\Omega k_{2} k_{3} |\boldsymbol k|^2 - (\beta^2/\eta) k_{1} k_{2}^{2} k_{3}}{4 \Omega^2 k_{3}^{2} |\boldsymbol k|^2 + (\beta^2/\eta)^2 k_{2}^{4}}\label{eq:M:1}\\
& \hat{M}_2(\boldsymbol k) = \frac{-2\Omega k_{1} k_{3} |\boldsymbol k|^2 - (\beta^2/\eta) k_{2}^{3}k_{3}}{4 \Omega^2 k_{3}^{2} |\boldsymbol k|^2 + (\beta^2/\eta)^2 k_{2}^{4}}\label{eq:M:2}\\
& \hat{M}_3(\boldsymbol k) = \frac{(\beta^2/\eta) k_{2}^{2}(k_{1}^{2} + k_{2}^{2})}{4 \Omega^2 k_{3}^{2} |\boldsymbol k|^2 + (\beta^2/\eta)^2 k_{2}^{4}}\label{eq:M:3}
\end{align}where the Fourier variable $\boldsymbol k \in {\mathbb Z}^d$ is such that $k_{3}\neq 0$. Note that $\hat{M}_{j}(\boldsymbol k)$ is not defined on $k_{3}=0$, since for the self-consistency of the model it is assumed that $\Theta$ and $\boldsymbol U$ have zero {\em vertical mean}. It can be directly checked that $k_{j} \cdot \hat{M}_j (\boldsymbol k) = 0$ for all $\boldsymbol k \in {\mathbb Z}^{d}\setminus \{ k_{3}=0\}$, and hence the velocity field $\boldsymbol U$ given by \eqref{eq:3} is indeed divergence-free. The physical parameters of the geodynamo are the following: $\Omega$ is the rotation rate of the Earth, $\eta$ is the magnetic diffusivity of the fluid core, $\kappa$ is the thermal diffusivity, and $\beta$ is the strength of the steady, uniform mean part of the magnetic field in the fluid core. The perturbation magnetic field vector $\boldsymbol{b}(t,x)$ is computed from $\Theta(t,\boldsymbol x)$ via the operator
\begin{align}
b_{j} =  (\beta/\eta) (-\Delta)^{-1} \partial_{2} M_{j} \Theta,\qquad \mbox{for all}\ j \in \{1,2,3\}.
\end{align}
We observe that the presence of the underlying magnetic field, reflected in the parameter $\beta^2/\eta$, produces a non-isotropic structure in the symbols $\hat{M}_j$.

It is important to note that although the symbols $\hat{M}_{i}$ are $0$-order homogenous under the isotropic scaling $\boldsymbol k \to \lambda \boldsymbol k$, due to their anisotropy the symbols $\hat{M}_i$ are {not} bounded functions of $\boldsymbol k$. To see this, note that whereas in the region of Fourier space where $k_{1} \leq \max \{ k_{2}, k_{3}\}$ the $\hat{M}_i$ are bounded by a constant, uniformly in $|\boldsymbol k|$, this is not the case on the ``curved'' frequency regions where $k_{3} = {\mathcal O}(1)$ and $k_{2} = {\mathcal O}(|k_{1}|^{r})$, with $0< r \leq 1/2$. In such regions the symbols are unbounded, since as $|k_{1}|\rightarrow \infty$ we have
\begin{align}
|\hat{M}_{1}(k_{1},|k_{1}|^{r},1)| \approx |k_{1}|^{r},\ |\hat{M}_{2}(k_{1},|k_{1}|^{r},1)| \approx |k_{1}|,\ |\hat{M}_{3}(k_{1},|k_{1}|^{r},1)| \approx |k_{1}|^{2r}. \label{eq:unbounded:symbol}
\end{align}
In fact, it may be shown that $|\hat{\boldsymbol M}(\boldsymbol k)| \leq C |\boldsymbol k|$, where $C(\beta,\eta,\Omega)>0$ is a fixed constant, and this bound is sharp. The fact that symbols are at most linearly \emph{unbounded} in the whole of Fourier space implies that $\boldsymbol M[\Theta]$ is the \emph{derivative} of a singular integral operator acting on $\Theta$ (see \cite{FriedlanderVicol11a} for details), as opposed to the case of the $(SQG)$ equation where $\boldsymbol U$ is the Riesz transform of $\Theta$.

In \cite{FriedlanderVicol11a} we studied a class of nonlinear active scalar equations of which the $(MG)$ system (\ref{eq:1})--(\ref{eq:M:3}) is a member, namely (\ref{eq:1}) with
\begin{align}
	U_j = \partial_j T_{ij}\Theta
\end{align}
for all $j \in \{ 1,2,3\}$ where $T_{ij}$ is a $3\times3$ matrix of Calder\'on-Zygmund operators of convolution type such that $\partial_i\partial_j T_{ij}\phi=0$ for any smooth function $\phi$. Inspired by the proof of Caffarelli and Vasseur \cite{CaffarelliVasseur10} for the global well-posedness of the critically diffusive $(SQG)$ equations, we used DeGiorgi techniques to obtain the global well-posedness of the critically diffusive $(MG)$ equations, namely (\ref{eq:1})--(\ref{eq:M:3}). We remark that  for both the diffusive $(MG)$ and $(SQG)$ equations {\em criticality}
is defined with respect to the $L^\infty$ maximum principle. In \cite{FriedlanderVicol11c} we considered the non-diffusive version of the $(MG)$ equations, namely (\ref{eq:1})--(\ref{eq:M:3}) with $\kappa=0$. In contrast with the critically diffusive problem where the $(MG)$ equation is globally well-posed and the solutions are $C^\infty$ smooth for positive time, we proved that when $\kappa=0$ the equation is Hadamard \emph{ill-posed} in any Sobolev space\footnote{The non-diffusive $(MG)$ equations are however locally well-posed in spaces of real-analytic functions, cf.~\cite{FriedlanderVicol11c}.}
 which embeds in $W^{1,4}$.

In this present paper we study the fractionally diffusive version of the $(MG)$ equation, namely
\begin{align}
	&\partial_t \Theta + \boldsymbol U \cdot \nabla \Theta = S - \kappa (-\Delta)^\gamma\Theta \label{eq:FractionalMG:1}\\
	& \nabla \cdot \boldsymbol U =0, \; \boldsymbol U = \boldsymbol M[\Theta] \label{eq:FractionalMG:2}
\end{align}
with $U_j = M_j[\Theta]$ where the symbols $\hat{M}_j$ are given by \eqref{eq:M:1}--\eqref{eq:M:3} and the parameter $\gamma \in (0,1)$. The main observation is that the structure of $\hat{M}_j$ (anisotropy, unboundedness, and evenness) and the {\em weak} smoothing effects of the nonlocal operator $(-\Delta)^\gamma$ combine to produce a sharp dichotomy across the value $\gamma=1/2$. More precisely, if $\gamma > 1/2$ the equations are locally well-posed, while if $\gamma < 1/2$ they are ill-posed in Sobolev spaces, in the sense of Hadamard.

In section \ref{sec:>1/2} we use energy estimates to prove that for $\gamma \in (1/2,1)$ the fractionally diffusive $(MG)$ equations are locally well-posed in Sobolev spaces $H^s({\mathbb T}^3)$ for $s > 5/2 + (1-2 \gamma)$. With an additional small data assumption we obtain a global well-posedness result. To see why one may not use energy estimates to obtain local well-posedness of the $(MG)$\  equations when $\gamma<1/2$, we point out that in the $H^{s}$ energy estimate for \eqref{eq:FractionalMG:1} there are only two terms for which more than ``$s$ derivatives'' fall on a single function:
\begin{align}
T_{bad} = \int \Lambda^{s} \boldsymbol U \cdot \nabla \Theta\; \Lambda^{s} \Theta \qquad \mbox{and} \qquad T_{good} = \int \boldsymbol U \cdot \nabla \Lambda^{s} \Theta\; \Lambda^{s} \Theta
\end{align}
where we denoted $\Lambda = (-\Delta)^{1/2}$. Since $\nabla \cdot \boldsymbol U = 0$, upon integrating by parts we have $T_{good} = 0$. On the other hand, the term $T_{bad}$ does not vanish in general. The diffusion competing with $T_{bad}$ is given by $\kappa \| \Lambda^{s+\gamma} \Theta \|_{L^{2}}^{2}$. Two issues arise. First, since $\boldsymbol U \approx \Lambda \Theta$ and $\gamma<1/2$, and one cannot control $\| \Lambda^{s-\gamma} \boldsymbol U \|_{L^{2}} $ with $\| \Lambda^{s+\gamma} \Theta \|_{L^{2}}$, and therefore bounding $T_{bad}$ could only be achieved by exploring some extra cancellation. Second, since the symbols $\hat{M}_{j}$ are even the operator $\boldsymbol M$ is {\em not} anti-symmetric, thus one {cannot} re-write $T_{bad}$ as $-\int \Lambda^{s} \Theta \cdot \boldsymbol M[\nabla \Theta\; \Lambda^{s}\Theta]$ (note that if one could do this, a suitable commutator estimate of Coifman-Meyer type would close the estimates at the level of Sobolev spaces, as in \cite{ChaeConstantinCordobaGancedoWu11,FriedlanderVicol11c}). Hence there is no a commutator structure in $T_{bad}$, and the estimates do not seem to close at the level of Sobolev spaces when $0< \gamma < 1/2$.

In section \ref{sec:<1/2} we consider the case $\gamma \in (0,1/2)$ and prove that the solution map associated with the Cauchy problem is {\em not} Lipschitz continuous with respect to perturbations in the initial data in the topology of a certain Sobolev space $X$. Hence the Cauchy problem is ill-posed in the sense of Hadamard. This is achieved by considering a specific steady profile $\Theta_0$, and constructing functions which are ``close'' to $\Theta_{0}$, but which in arbitrarily short time deviate arbitrarily far from $\Theta_{0}$, when measured in $X$. The arguments hinge on using techniques of continued fractions to exhibit an unstable eigenvalue for the linearized equation (cf. \cite{FriedlanderStraussVishik97,MeshalkinSinai61}). These techniques produce a \emph{lower} bound on the growth rate of eigenvalues to the linearized equations, and in the case where $\gamma \in (0,1/2)$ we prove that the magnitude of this lower bound can be made arbitrarily large. Once unstable eigenvalues of arbitrarily large part have been obtained for the linearized equations, the Lipschitz ill-posednedness of the full non-linear equations may be obtained using classical arguments (see, e.g.~\cite{FriedlanderVicol11c,GerardVaretDormy10,GuoNguyen10,Renardy09,Tao06}). We emphasize that the mechanism producing ill-posedness is not merely the order one derivative loss in the map $\Theta \to \boldsymbol U$. Rather, it is the combination of the derivative loss with the anisotropy of the symbol $M$ and the fact that this symbol is \emph{even}. We note that the even nature of the symbol of $M$ plays a central role in the proof of the non-uniqueness given in \cite{Shvydkoy11} for the $L^\infty$-weak solutions of the non-diffusive $(MG)$ equations.

In section \ref{sec:=1/2} we study the more subtle transitional case when $\gamma=1/2$. In this case, if the initial data is ``small'' with respect to $\kappa$, we use energy estimates to prove that there exists a unique global solution. In dramatic contrast, there exists initial data which is ``large'' with respect to $\kappa$, and for which the associated linear operator has arbitrarily large unstable eigenvalues, which may be use to prove that the equations are Hadamard ill-posed. Situations where the qualitative behavior of the solution depends crucially on the norm of initial data are also encountered for other evolution equations. For example, in the two-dimensional Keller-Segel model of chemotaxis, properties of the solution are strongly dependent on the total mass $m$ of cells. If $m < 8\pi$, a global bounded solution of the initial value problem exists and is dispersed for $t\to \infty$, while for $m > 8\pi$, blow-up in finite time occurs. The critical case $m=8\pi$ is by now understood and the result states that a global solution exists and possibly becomes unbounded as $t \to \infty$ (see~\cite{HillenPainter09} and references therein). We also mention that for the Rayleigh-Taylor and the Helmholtz problems it is ``geometric'' conditions, rather than merely the size of the data, which leads to ill-posedness (see, for instance \cite{Ebin88}).

In section \ref{sec:well-prepared} we take advantage of the anisotropy of the symbols \eqref{eq:M:1}--\eqref{eq:M:3} and observe an interesting phenomenon: when the initial data and the source term are ``well-prepared'' in an appropriate sense, it is possible to prove stronger regularity results for the ensuing solutions. More specifically, when the initial data and the force have ``thin'' Fourier support, i.e. they are supported purely on a plane $\mathcal{P}_q= \{ \boldsymbol k=(k_1,k_2,k_3) \in {\mathbb Z}^3: k_2= q k_1\} $ in frequency space,  where $q$ is a fixed non-zero rational number, the operator $\boldsymbol M$ behaves like an operator of order zero. The reason is that the Fourier symbols $\hat{M}_j$ are unbounded only on curved regions in frequency space (cf.~\eqref{eq:unbounded:symbol} above), but if $\boldsymbol k \in \mathcal P_q$ the $\hat{M}_j(\boldsymbol k)$ are bounded by a constant depending on $q$. This observation, combined with the fact that when the data and source have frequency support on $\mathcal P_q$, then so does the solution of the $(MG)$\  equations at all later times, means that the smoothing properties of the fractional diffusion term are stronger than they are for the generic data situation. Hence it is possible to prove stronger regularity results, and in particular the local existence and uniqueness of Sobolev solutions holds {\em for all} values of $\gamma \in (0,1)$.
We note that this is not in contradiction with our results proven in sections~\ref{sec:<1/2} and \ref{sec:=1/2}, since in order to obtain ill-posedness we need to send $q \to 0$ (so that we obtain eigenvalues of the linear operator with arbitrarily large real part), whereas in the case of well-prepared data the value of $q>0$ is {\em fixed}. Other uses of thin Fourier support for different problems can be found, e.g. in \cite{BernicotGermain09,Tao06}.

\section{Preliminaries}

This section contains a few auxiliary results used in the paper. In particular, we recall the, by now classical, product and commutator estimates, as well as the Sobolev embedding inequalities. Proofs of these results can be found for instance in \cite{KenigPonceVega91,Stein93,Tao06}. Let us denote $\Lambda = (-\Delta)^{1/2}$.

\begin{proposition}[{\bf Product estimate}]
\label{prop:calculus}
If $s > 0$, then 	for all $f,g \in H^s\cap L^\infty$, and we have the estimates
	\begin{align}
		&\|\Lambda^s(fg)\|_{L^p} \leq C\left( \|f\|_{L^{p_1}} \|\Lambda^s g\|_{L^{p_2}} + \|\Lambda^sf\|_{L^{p_3}} \|g\|_{L^{p_4}}\right),
	\end{align}
	where $1/p = {1}/{p_1}+{1}/{p_2}= {1}/{p_3}+ {1}/{p_4}$, and $p,p_{2},p_{3} \in(1,\infty)$. In particular
	\begin{align}
		&\|\Lambda^s(fg)\|_{L^2} \leq C\left( \|f\|_{L^{\infty}} \|\Lambda^s g\|_{L^{2}} + \|\Lambda^sf\|_{L^{2}} \|g\|_{L^{\infty}}\right). \label{eq:prop:product}
	\end{align}
\end{proposition}

In the case of a commutator we have the following estimate.

\begin{proposition}[{\bf Commutator estimate}]
	Suppose that $s >0$ and $p \in (1,\infty)$. If, $f,g \in \mathcal{S}$, then
	\begin{align}
		&\|\Lambda^s(fg) - f \Lambda^{s} g \|_{L^p} \leq C\left(\| \nabla f\|_{L^{p_1}}\|\Lambda^{s-1} g\|_{L^{p_2}} + \|\Lambda^s f\|_{L^{p_3}}\|g\|_{L^{p_4}} \right) \label{eq:prop:commutator}
	\end{align}
	where $s>0$, ${1}/{p} = {1}/{p_1}+{1}/{p_2}={1}/{p_3}+{1}/{p_4}$, and $p,p_{2},p_{3} \in(1,\infty)$.
\end{proposition}

We shall use as well the following Sobolev inequality.

\begin{proposition}[{\bf Sobolev inequality}] \label{prop:Sobolev}
Suppose that $q > 1$, $p \in [q,\infty)$ and ${1}/{p}={1}/{q}-{s}/{d}$. Suppose that $\Lambda^s f \in L^q$, then $f \in L^p$ and there is a constant $C > 0$ such that
	\begin{align}
		\|f\|_{L^p} \leq C \|\Lambda^s f\|_{L^q}.
	\end{align}
\end{proposition}

Throughout this paper we shall make use the following definition of solutions of \eqref{eq:1}--\eqref{eq:3}.
\begin{definition}[\bf Solution of the $(MG)$\ \ equation]
 Let $s \geq 1/2$, $T>0$, $\kappa>0$, and $\gamma \in (0,1)$. Given $\Theta_0 \in H^s({\mathbb T}^3)$, we call a function
 \begin{align}
\Theta \in L^\infty(0,T;H^s({\mathbb T}^3)) \cap L^2(0,T; H^{s+\gamma}({\mathbb T}^3))
\end{align}
a {\em solution} of the Cauchy problem \eqref{eq:1}--\eqref{eq:3}, if
\begin{align}
\int_{{\mathbb T}^3} \left( \Theta(t,\cdot) - \Theta_0 \right)  \varphi d\boldsymbol x  + \kappa \int_0^t  \int_{{\mathbb T}^3} \Theta (-\Delta)^\gamma \varphi d\boldsymbol x ds - \int_0^t \int_{{\mathbb T}^3} \Theta\; \boldsymbol U \cdot \nabla \varphi d\boldsymbol x ds = 0
\end{align}
holds for all test functions $\varphi \in C_0^\infty({\mathbb T}^3)$, and all $t\in (0,T)$.
\end{definition}

\section{The regime $1/2<\gamma < 1$: well-posedness results} \label{sec:>1/2}

It has been shown in \cite{FriedlanderVicol11a} that the system (\ref{eq:1})--(\ref{eq:M:3}) is globally well-posed when $\gamma =1$. In this section we prove that in the range $\gamma \in (1/2,1)$ equations (\ref{eq:1})--(\ref{eq:M:3}) are locally well-posed in $H^s({\mathbb T}^3)$. With an additional assumption on the size of the initial data and the source term, we obtain a global well-posedness result.
\footnote{For our convenience, we choose to work in sub-critical Sobolev spaces. The following proofs can be also carried out in the setting of critical Besov spaces $B^s_{2,1}$ or in general $B^{s}_{p,q}$.}

Regarding arbitrarily large initial data, we obtain the following result.

\begin{theorem}[{\bf Local existence}]\label{thm:121:local}
Let $\gamma \in (1/2,1)$, and fix $s>5/2 + (1-2\gamma)$. Assume that $\Theta_0 \in H^s({\mathbb T}^3)$ and $S \in L^\infty(0,\infty;H^{s-\gamma}({\mathbb T}^3))$ have zero-mean on ${\mathbb T}^{3}$. Then there exists a time $T>0$ and a unique smooth solution
\begin{align} \Theta \in L^\infty(0,T;H^s({\mathbb T}^3)) \cap L^2(0,T;H^{s+\gamma}({\mathbb T}^3))
\end{align} of the Cauchy problem \eqref{eq:FractionalMG:1}--\eqref{eq:FractionalMG:2}.
\end{theorem}
\begin{proof}
	 We multiply equation (\ref{eq:1}) by $\Lambda^{2s}\Theta$, integrate by parts to obtain the following $H^s$-energy inequality
\begin{align}\label{eq:p1}
&\frac{1}{2}\frac{d}{dt}\|\Lambda^s\Theta\|^2_{L^2} + \kappa\|\Lambda^{s+\gamma}\Theta\|^2_{L^2} \leq \left| \int_{{\mathbb R}^3} S \Lambda^{2s}\Theta\right| + \left| \int_{{\mathbb R}^3} \boldsymbol U \cdot \nabla \Theta \Lambda^{2s}\Theta\right|.
\end{align}
We estimate the first term on the right side by
\begin{align}
	&\left| \int_{{\mathbb R}^3} S \Lambda^{2s}\Theta\right| \leq \|\Lambda^{s-\gamma}S\|_{L^2}\|\Lambda^{s+\gamma}\Theta\|_{L^2} \leq \frac{1}{2\kappa} \|\Lambda^{s-\gamma}S\|_{L^2}^2 + \frac{\kappa}{2}\|\Lambda^{s+\gamma}\Theta\|_{L^2}^2.
\end{align}
To handle the second term we proceed as follows. First note that
\begin{align}
	& \left| \int_{{\mathbb R}^3} \boldsymbol U \cdot \nabla \Theta \Lambda^{2s}\Theta\right| = \left| \int_{{\mathbb R}^3} \Lambda^{s-\gamma}(\boldsymbol U\cdot\nabla \Theta)\Lambda^{s+\gamma}\Theta\right| \leq \|\Lambda^{s-\gamma}(\boldsymbol U \cdot \nabla \Theta)\|_{L^2}\|\Lambda^{s+\gamma}\Theta\|_{L^2}.
\end{align}
The estimate of the product term follows from Proposition \ref{prop:calculus}. Hence, we have
\begin{align}\label{eq:p8}
	&\|\Lambda^{s-\gamma}(\boldsymbol U \cdot \nabla \Theta)\|_{L^2} \leq C(\|\Lambda^{s-\gamma}\boldsymbol U\|_{L^p}\|\nabla \Theta\|_{L^{\frac{2p}{p-2}}} + \|\Lambda^{s-\gamma}\nabla \Theta\|_{L^p}\|\boldsymbol U\|_{L^{\frac{2p}{p-2}}})
\end{align}
for some $2 < p < \infty$ to be chosen later.
Recall that $U_j = M_j\Theta $, where the symbol of the multiplier $M_j$ enjoys a uniform bound $|\hat{M}_j(\boldsymbol k)| \leq C|\boldsymbol k|$, for $j=1,2,3$. Therefore $\Lambda^{-1} M_{j}$ are bounded operators on $L^{p}$ (see \cite[Section 4]{FriedlanderVicol11a} for details) and we obtain
\begin{align}
	&\|\Lambda^{s-\gamma}\boldsymbol U\|_{L^p} \leq C \|\Lambda^{s-\gamma+1}\Theta\|_{L^p}
\end{align}
and
\begin{align}
\| \boldsymbol U \|_{L^{\frac{2p}{p-2}}} \leq C \| \Lambda \Theta \|_{L^{\frac{2p}{p-2}}}.
\end{align}
Hence, the product term is bounded by
\begin{align}\label{eq:2.8}
	&\|\Lambda^{s-\gamma}(\boldsymbol U \cdot \nabla \Theta)\|_{L^2} \leq C \| \Lambda \Theta\|_{L^{\frac{2p}{p-2}}} \|\Lambda^{s-\gamma+1}\Theta\|_{L^p}.
\end{align}
We now fix an arbitrary $p$  such that
\begin{align*}
\frac{3}{s-1} < p  <  \frac{6}{2(1-2\gamma)+3} = \frac{6}{5-4\gamma}.
\end{align*}
Note that $p>2$ since $s>3/2$, and the range for $p$ is non-empty since $s> 5/2 + (1-2\gamma)$. For $\gamma \in (1/2,1)$, our choice of $p$ and the Sobolev embedding (Proposition~\ref{prop:Sobolev}) gives
\begin{align}
\|\Lambda^{s-\gamma+1}\Theta\|_{L^p} \leq \|\Lambda^s\Theta\|^{1-\zeta}_{L^2}\|\Lambda^{s+\gamma}\Theta\|_{L^2}^{\zeta}
\end{align}
where $\zeta \in (0,1)$ may be computed explicitly from $\gamma \zeta = 3/2 - 3/p + 1-\gamma$. In addition, since $\Theta$ has zero mean, and $p> 3/(s-1)$, from the Sobolev embedding we obtain
\begin{align}\label{eq:2.9}
	&\|\Lambda \Theta\|_{L^{\frac{2p}{p-2}}} \leq C \|\Lambda^s\Theta\|_{L^2}.
\end{align}
Combining estimates (\ref{eq:p1})-(\ref{eq:2.9}) gives
\begin{align}\label{eq:p6}
	& \frac{d}{dt}\|\Lambda^s\Theta\|^2_{L^2} + \kappa\|\Lambda^{s+\gamma}\Theta\|^2_{L^2} \leq \frac{1}{\kappa} \|\Lambda^{s-\gamma}S\|_{L^2}^{2} + C\|\Lambda^s\Theta\|_{L^2}^{2-\zeta}\|\Lambda^{s+\gamma}\Theta\|_{L^2}^{1+\zeta}
\end{align}
where $ 0 < \zeta< 1$ is as defined earlier.
The second term on the right side of (\ref{eq:p6}) is bounded using the $\epsilon$-Young inequality as
\begin{align}
 \frac{\kappa}{2}\|\Lambda^{s+\gamma}\Theta\|_{L^2}^2 +C \kappa^{- \frac{2(1+\zeta)}{1-\zeta}} \|\Lambda^s\Theta\|_{L^2}^{\frac{2 (2-\zeta)}{1-\zeta}}.
\end{align}
and we finally obtain the following estimate
\begin{align}\label{eq:p7}
	&\frac{d}{dt}\|\Lambda^s\Theta\|^2_{L^2} + \frac{\kappa}{2}\|\Lambda^{s+\gamma}\Theta\|^2_{L^2} \leq \frac{1}{\kappa}\|\Lambda^{s-\gamma}S\|_{L^2}^2 + C \kappa^{- \frac{2(1+\zeta)}{1-\zeta}} \|\Lambda^s\Theta\|_{L^2}^{\frac{2 (2-\zeta)}{1-\zeta}}.
\end{align}
Using Gronwall's inequality, from estimate (\ref{eq:p7}) we may deduce the existence of a positive time $T=T(\|S\|_{L^\infty(0,T;H^{s-\gamma})},\|\Theta_0\|_{H^s},\kappa)$ such that $\theta \in L^\infty(0,T;H^s({\mathbb T}^3))\cap L^2(0,T;H^{s+\gamma}({\mathbb T}^3))$. Note that we have $2(2-\zeta)/(1-\zeta) >2$, and hence we may not obtain the global existence of solutions from the energy estimate \eqref{eq:p7}, if the initial data has large $H^{s}$ norm. These a priori estimates can be made formal using a standard approximation procedure. We omit further details.
\end{proof}

The second main result of this section concerns global well-posedness in case of small initial data.

\begin{theorem}[{\bf Global existence for small data}]\label{thm:121:global}
Let $\gamma$ and $S$ be as in the statement of Theorem~\ref{thm:121:local}, and let $ \Theta_0 \in H^{s}({\mathbb T}^3)$ have zero-mean on ${\mathbb T}^3$, where $s > 5/2+(1-2\gamma)$. There exists a  small enough constant $\varepsilon>0$ depending on $\kappa$, such that if  $\Vert\Theta_0\Vert_{L^2}^\alpha \Vert \Theta_{0} \Vert_{H^{s}}^{1-\alpha} +\Vert\Theta_0\Vert_{L^2}^\alpha \Vert S \Vert_{L^\infty(0,\infty;H^{s-\gamma})}^{1-\alpha}\leq \varepsilon$, where $\alpha = 1 - (5/2+1-2\gamma)/s$, then the unique smooth solution $\Theta$ of the Cauchy problem  \eqref{eq:FractionalMG:1}--\eqref{eq:FractionalMG:2} is global in time, i.e.
\begin{align} \Theta \in L^\infty(0,\infty;H^s({\mathbb T}^3)).
\end{align}
\end{theorem}
\begin{proof}
We proceed as in the proof of Theorem \ref{thm:121:local}. The product term in (\ref{eq:p8}) is now estimated by
\begin{align}
	&\|\Lambda^{s-\gamma}(\boldsymbol U \cdot \nabla \Theta)\|_{L^2} \leq C(\|\Lambda^{s-\gamma}\boldsymbol U\|_{L^p}\|\nabla \Theta\|_{L^{\frac{2p}{p-2}}}+ \|\Lambda^{s-\gamma}\nabla \Theta\|_{L^p}\|\boldsymbol U\|_{L^\frac{2p}{p-2}}),
\end{align}
where
\begin{align*}
	p = \frac{6}{2(1-2\gamma)+3}
\end{align*}
so that
\begin{align}
	&\|\Lambda^{s+1-\gamma}\Theta\|_{L^p} \leq C\|\Lambda^{s+\gamma}\Theta\|_{L^2}.
\end{align}
With this choice of $p$ and the above embedding, the product estimate gives us
\begin{align}\label{eq:p9}
	\|\Lambda^{s-\gamma}(\boldsymbol U \cdot \nabla \Theta)\|_{L^2} \leq C\|\nabla \Theta\|_{L^{\frac{2p}{p-2}}}\|\Lambda^{s+\gamma} \Theta\|_{L^2}.
\end{align}
Combining (\ref{eq:p9}) with (\ref{eq:p1}) and proceeding as in (\ref{eq:p7}) we obtain
\begin{align}\label{eq:p10}
	&\frac{1}{2}\frac{d}{dt}\|\Lambda^s\Theta\|^2_{L^2} + \kappa\|\Lambda^{s+\gamma}\Theta\|^2_{L^2} \leq  \|\Lambda^{s-\gamma}S\|_{L^2}\|\Lambda^{s+\gamma}\Theta\|_{L^2} + C\|\nabla \Theta\|_{L^{\frac{2p}{p-2}}}\|\Lambda^{s+\gamma} \Theta\|_{L^2}^2,
\end{align}
which in turn implies
\begin{align}\label{eq:p11}
	&\frac{1}{2}\frac{d}{dt}\|\Lambda^s\Theta\|^2_{L^2} + \frac{\kappa}{2}\|\Lambda^{s+\gamma}\Theta\|^2_{L^2} \leq \frac{1}{2\kappa}\|\Lambda^{s-\gamma}S\|_{L^2}^2 + C\|\nabla \Theta\|_{L^{\frac{2p}{p-2}}}\|\Lambda^{s+\gamma} \Theta\|_{L^2}^2.
\end{align}
Observe that
\begin{align}\label{eq:p12}
	\|\nabla \Theta\|_{L^{2p/(p-2)}} \leq C \|\Theta\|_{L^2}^{\alpha}\|\Lambda^s \Theta\|_{L^2}^{1-\alpha},
\end{align}
where $\alpha = 1- (5/2+1-2\gamma)/s$.
Therefore, if
\begin{align}\label{eq:p13}
	\|\Theta\|_{L^2}^{\alpha}\|\Lambda^s \Theta\|_{L^2}^{1-\alpha} \leq \frac{\kappa}{4C},
\end{align}
estimate (\ref{eq:p11}), combined with the Poincar\'e inequality $\| \Lambda^{s+\gamma} \Theta \|_{L^{2}} \geq \| \Lambda^{s} \Theta \|_{L^{2}}$, shows that
\begin{align}
\frac{d}{dt} \|\Lambda^s\Theta\|^2_{L^2} + \frac{\kappa}{2}\|\Lambda^{s}\Theta\|^2_{L^2} \leq \frac{1}{\kappa} \| S\|_{L^{\infty}_{t}H^{s-\gamma}_{x}}^{2} .
\end{align}
and hence
\begin{align} \label{eq:p13a}
 \|\Lambda^s\Theta(t,\cdot) \|^2_{L^2}  \leq \|\Lambda^s\Theta_{0} \|^2_{L^2} + \frac{2}{\kappa^{2}} \| S\|_{L^{\infty}_{t}H^{s-\gamma}_{x}}^{2}
\end{align}
for all $t>0$. Note that taking the $L^2$-product of the equation with $\Theta$ gives for any $t >0$
\begin{align}\label{eq:p14}
	\|\Theta(t,\cdot)\|^2_{L^2} + \kappa \int_0^t\|\Lambda^\gamma \Theta\|^2_{L^2}\;d\tau \leq \|\Theta_0\|^2_{L^2},
\end{align}
which gives us a basic uniform estimate of $\Theta$ in $L^{\infty}_{t}L^2_{x}$. Hence, from \eqref{eq:p13a} and \eqref{eq:p14} we obtain that condition \eqref{eq:p13} is satisfied for all $t>0$ as long as we have
\begin{align}\label{eq:p15}
 	& \|\Theta_0\|_{L^2}^{\alpha}\|\Lambda^s \Theta_0\|_{L^2}^{1-\alpha} + \|\Theta_0\|_{L^2}^{\alpha} \|\Lambda^{s-\gamma}S(\tau,\cdot)\|_{L^\infty(0,\infty;L^2)}^{1-\alpha} < \epsilon,
\end{align}
where $\epsilon$ is sufficiently small, thereby concluding the proof of the theorem.
\end{proof}

\section{The regime $0<\gamma < 1/2$: ill-posedness results} \label{sec:<1/2}

We now turn to the situation when $\gamma \in (0,1/2)$, $\kappa > 0$, and prove that there exists an example of an initial datum and time independent force for which it is possible to prove that the fractionally diffusive $(MG)$\  equation is ill-posed in the sense of Hadamard. The arguments by which we prove this follow the lines of the proof in the case when $\kappa = 0$ in \cite{FriedlanderVicol11c}. We sketch here the proof of $\gamma \in (0,1/2)$, but as we shall prove below in Section~\ref{sec:=1/2_large_data}, the result is also true when $\gamma = 1/2$ and the initial data is {\em large} in a suitable sense.

\subsection{Linear ill-posedness in $L^2$} \label{eq:sec:linear:ill}
The classical approach to Hadamard ill-posedness for non-linear problems is to first linearize the equations about a suitable steady state $\Theta_0$, and then prove that the resulting linear operator $L$ has eigenvalues with arbitrarily large real part in the unstable region (see for instance~\cite{FriedlanderVicol11c,GerardVaretDormy10,GuoNguyen10,Renardy09}, and references therein).
Let $\Theta_{0} = a \sin (mx_{3})$ for some positive amplitude $a$ and integer frequency $m \geq 1$,  and define $S = \kappa a m^{2\gamma} \sin (m x_{3})$. It is not hard to verify that $\Theta_{0}$ is indeed a steady state of \eqref{eq:1}--\eqref{eq:2}. We consider the {\em linear} evolution of the perturbation $\theta = \Theta - \Theta_{0}$, obtained from linearizing \eqref{eq:1}--\eqref{eq:2}
\begin{align}
\partial_{t} \theta + a m \cos(m x_{3}) M_{3}\theta+ \kappa (-\Delta)^{\gamma} \theta =  0 \label{eq:SN:5.1},
\end{align}
where we recall that $M_{3}$ is the Fourier multiplier with symbol defined in \eqref{eq:M:3}.
The following theorem states that the linear equation \eqref{eq:SN:5.1} is Hadamard Lipschitz ill-posed in $L^{2}$.
\begin{theorem}[\bf Linear ill-posedness] \label{thm:SN:5.1}
The Cauchy problem associated to the linear evolution
\begin{align}
  \partial_t \theta = L\theta \label{eq:linearequation}
\end{align} where the linear operator $L$ and the steady state $\Theta_{0}$ are given by
\begin{align}
L\theta(\boldsymbol x,t) &= - M_3 \theta (\boldsymbol x,t)\; \partial_3 \Theta_0(x_3) - \kappa (-\Delta)^{\gamma} \theta\label{eq:L:def}\\
\Theta_0(x_3) & = a \sin(m x_3) \label{eq:steadystatedef}
\end{align}for some $a>0$,  integer $m\geq 1$, and $\gamma \in (0,1/2]$, is ill-posed in the sense of Hadamard over $L^{2}$. More precisely, for any $T>0$ and any $K>0$, there exists a real-analytic initial data $\theta(0,\boldsymbol x)$ such that the Cauchy problem associated to \eqref{eq:linearequation}--\eqref{eq:steadystatedef} has no solution $\theta \in L^{\infty}(0,T;L^{2})$ satisfying
\begin{align}
\sup_{t\in [0,T)} \Vert \theta(t,\cdot) \Vert_{L^{2}} \leq K \Vert \theta(0,\cdot)\Vert_{Y} \label{eq:thm:boundedness}
\end{align}
where $Y$ is any Sobolev space embedded in $L^{2}$.
\end{theorem}
As we mentioned above, in order to prove Theorem~\ref{thm:SN:5.1} we construct a sequence of eigenvalues $\sigma^{(j)}$ of the operator $L$, which diverge to $\infty$ as $j \rightarrow \infty$.
For this purpose, given any {\em fixed} integer $j \geq 1$ we seek a solution $\theta$ to \eqref{eq:linearequation} of the form
\begin{align}
\theta(t,\boldsymbol x) = e^{\sigma t} \sin(j^{2} x_{1}) \sin(j x_{2}) \sum_{p\geq 1} c_{p} \sin(m p x_{3}) \label{eq:SN:5.2}
\end{align}
where the sequence $c_{p}$ decays rapidly as $p \rightarrow \infty$. We shall construct such a solution $\theta$ with $\sigma$ real and positive, and in addition obtain bounds on the value of $\sigma$. Inserting \eqref{eq:SN:5.2} into \eqref{eq:SN:5.1} and using the definition of $M_{3}$, we obtain
\begin{align}
\sigma_p \sum_{p\geq 1} c_{p} \sin(m p x_{3}) + \sum_{p\geq 1} \frac{c_{p}}{\alpha_{p}} \left( \sin ( m (p+1) x_{3}) + \sin ( m (p-1) x_{3}) \right) = 0, \label{eq:SN:5.3}
\end{align}
where we have denoted
\begin{align}
\sigma_{p} = \sigma + \kappa (j^{4} + j^{2} + (mp)^{2})^{\gamma} \label{eq:SN:5.6}
\end{align}
and
\begin{align}
\alpha_{p} = \frac{2^{3} \Omega^{2} (mp)^{2} (j^{4} + j^{2} + (mp)^{2}) + 2 \mu^{2} j^{4}}{a \mu m j^{2} (j^{4} + j^{2})} \label{eq:SN:5.4}
\end{align}
for any integer $p\geq 1$ (note that $j$ is fixed). Here  $\mu = \beta^{2}/\eta$. An essential feature of the coefficients $\alpha_{p}$ is that they grow rapidly with $p$ as $p\rightarrow \infty$. Equation \eqref{eq:SN:5.4} gives the recurrence relation for the unknown coefficients $c_{p}$:
\begin{align}
&\sigma_{p} c_{p} + \frac{c_{p+1}}{\alpha_{p+1}} + \frac{c_{p-1}}{\alpha_{p-1}} = 0, \mbox{ for } p\geq 2 \label{eq:SN:5.7}\\
&\sigma_{1} c_{1} + \frac{c_{2}}{\alpha_{2}} = 0, \mbox{ for } p = 1. \label{eq:SN:5.8}
\end{align}
Note that given any $\sigma>0$ (which then uniquely defines $\sigma_{p}$ for all $p\geq 1$ cf.~\eqref{eq:SN:5.6}) and given any $c_{1} > 0$, one may use the recursion relations \eqref{eq:SN:5.7}--\eqref{eq:SN:5.8} to solve for all $c_{p}$ with $p \geq 2$. However, only for suitable values of $\sigma$ do the $c_{p}$'s vanish sufficiently fast so that $\theta$ is $C^{\infty}$ smooth. In this direction we prove:
\begin{lemma} \label{lemma:SN:5.2}
 Let $\alpha_{p}$ be defined by \eqref{eq:SN:5.4}, where the positive integers $m$ and $j$ are fixed, and $\kappa, \mu, \Omega, a$ are fixed physical parameters. Also, define $c_{1} = \alpha_{1}$. Then there exists a real eigenvalue $\sigma = \sigma^{(j)}>0$, and rapidly decaying sequence $\{c_{p}\}_{p\geq 2}$ which satisfies \eqref{eq:SN:5.7}--\eqref{eq:SN:5.8}. Furthermore the lower bound
 \begin{align}
\sigma^{(j)} > \frac{a \mu m j^{2} (j^{4} + j^{2})}{2^{3} \Omega^{2} m^{2} (j^{4} + j^{2} + 4 m^{2}) + 2 \mu^{2} j^{4} } - \kappa (j^{4} + j^{2} + 4 m^{2})^{\gamma}
\end{align}
holds, and $c_{p} = \mathcal O(C^{p}/ (p-1)!^{4})$ as $p \rightarrow \infty$, for some constant $C$ which depends on $\sigma^{(j)},j$ and the physical parameters.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:SN:5.2}]
 We define
 \begin{align}
\eta_{p} = \frac{c_{p} \alpha_{p-1}}{c_{p-1} \alpha_{p}} \label{eq:SN:5.9}
\end{align}
and write \eqref{eq:SN:5.7}--\eqref{eq:SN:5.8} in the form
\begin{align}
& \sigma_{p} \alpha_{p} + \eta_{p+1} + \frac{1}{\eta_{p}} = 0, \qquad p \geq 2\label{eq:SN:5.10}\\
& \sigma_{1} \alpha_{1} + \eta_{2} =0, \qquad p=1 \label{eq:SN:5.11}.
\end{align}
Using \eqref{eq:SN:5.10} to write $\eta_{2}$ as an infinite continued fraction and equating with $\eta_{2}$ given by \eqref{eq:SN:5.11} gives the characteristic equation
\begin{align}
\sigma_{1} \alpha_{1} = \frac{1}{ \sigma_{2} \alpha_{2} - \frac{1}{ \sigma_{3} \alpha_{3} - \frac{1}{ \sigma_{4} \alpha_{4} - {\dots}} } }  \label{eq:SN:5.12}.
\end{align}
Recalling the definition of $\sigma_{p}$ for $p\geq 1$ cf.~\eqref{eq:SN:5.6}, we observe that \eqref{eq:SN:5.12} is an equation with only one unknown, namely $\sigma$.
For real values of $\sigma$ we define the infinite continued fraction $F_{p}(\sigma)$ by
\begin{align}
F_{p}(\sigma) = \frac{1}{\sigma_{p} \alpha_{p} - \frac{1}{\sigma_{p+1} \alpha_{p+1} - \frac{1}{\sigma_{p+2} \alpha_{p+2} - \dots}}} \label{eq:SN:5.13}
\end{align}
and the
function
\begin{align}
G_{p}(\sigma) = \frac{\sigma_{p} \alpha_{p} - \sqrt{\sigma_{p}^{2} \alpha_{p}^{2} - 4}}{2} = \frac{2}{\sigma_{p} \alpha_{p} + \sqrt{\sigma_{p}^{2} \alpha_{p}^{2} -4 }} \label{eq:SN:5.14}
\end{align}
for all $p\geq 2$, and all $\sigma$ such that $\sigma_{2} \alpha_{2}> 2$. We note that due to the very rapid growth of the $\alpha_{p}$'s, for real $\sigma$ the continued fraction $F_{p}$ defines a function which is smooth except for a set of points such with $\sigma_{2} \alpha_{2} <2$. For the rest of the proof we will always assume that $\sigma$ is real such that $\sigma_{2} \alpha_{2} >  2$, which also implies that $\sigma_{p} \alpha_{p}>2$ for all $p\geq 2$.

We note that by construction $G_{p}$ satisfies
\begin{align}
G_{p}(\sigma) = \frac{1}{\sigma_{p} \alpha_{p} - G_{p}(\sigma)} = \frac{1}{\sigma_{p} \alpha_{p} - \frac{1}{\sigma_{p} \alpha_{p} - \frac{1}{\sigma_{p} \alpha_{p} - \dots}}}. \label{eq:SN:5.15}
\end{align}
Since we have $\sigma_{p} \alpha_{p} \rightarrow \infty$ as $p \rightarrow \infty$, we have that $G_{p}(\sigma) \rightarrow 0$ as $p \rightarrow \infty$ for every fixed $\sigma$.
Also, we clearly have
\begin{align}
G_{2}(\sigma) > G_{3}(\sigma) > G_{4}(\sigma) > \ldots \geq 0 \label{eq:SN:5.16}
\end{align}
and
\begin{align}
G_{p+1}(\sigma) < G_{p}(\sigma) < \sigma_{p} \alpha_{p} \label{eq:SN:5.17}
\end{align}
for all $p\geq 2$. Hence, from \eqref{eq:SN:5.15} and \eqref{eq:SN:5.16} we obtain
\begin{align}
G_{2}(\sigma) > \frac{1}{\sigma_{2} \alpha_{2} - G_{3}(\sigma)} > \frac{1}{\sigma_{2} \alpha_{2} - \frac{1}{\sigma_{3} \alpha_{3} - G_{4}(\sigma)}} > 0 \label{eq:SN:5.18}.
\end{align}
An inductive argument then gives
\begin{align}
G_{2}(\sigma) >\frac{1}{\sigma_{2} \alpha_{2} - \frac{1}{\sigma_{3} \alpha_{3} - \frac{1}{\sigma_{4} \alpha_{4} - \dots} }} = F_{2}(\sigma) \geq 0 \label{eq:SN:5.19}
\end{align}
for all $\sigma$ such that $\sigma_{2} \alpha_{2} > 2$. Repeating this constructive argument we obtain that $G_{3}(\sigma) > F_{3}(\sigma) \geq 0$, and hence
\begin{align}
0< \sigma_{2} \alpha_{2} - G_{3}(\sigma) < \sigma_{2} \alpha_{2} - F_{3}(\sigma) < \sigma_{2} \alpha_{2}
\end{align}
so that
\begin{align}
F_{2}(\sigma)  = \frac{1}{\sigma_{2} \alpha_{2} - F_{3}(\sigma)} > \frac{1}{\alpha_{2} \sigma_{2}} \label{eq:SN:5.20}.
\end{align}

In summary, we have proven that $1/(\sigma_{2} \alpha_{2}) < F_{2}(\sigma) < G_{2}(\sigma)$ for all $\sigma$ such that $\sigma_{2} \alpha_{2} > 2$, and the straight line $\sigma_{1}\alpha_{1}$ intersects both the graph of $1/(\sigma_{2} \alpha_{2})$ and the graph of $G_{2}(\sigma)$ in this rage of $\sigma$. Hence, it follows that there exists a real solution $\sigma^{(j)}$ of the equation
\begin{align}
F_{2}(\sigma^{(j)}) = \sigma_{1}^{(j)} \alpha_{1} \label{eq:SN:5.21},
\end{align}
with $\sigma_{2}^{(j)} \alpha_{2} > 2$, where we denote $\sigma_{p}^{(j)} = \sigma^{(j)} + \kappa (j^{4} + j^{2} + m^{2} p^{2})^{\gamma}$, for all $p\geq 2$. That is, $\sigma^{(j)}$ is a real positive solution of the characteristic equation \eqref{eq:SN:5.12}. Furthermore, due to \eqref{eq:SN:5.19} and \eqref{eq:SN:5.20} after a short calculation we obtain an upper and a lower bound on $\sigma^{(j)}$, namely
\begin{align}
\frac{1}{\alpha_{1} \alpha_{2}} < \sigma_{1}^{(j)} \sigma_{2}^{(j)} < \frac{2}{\alpha_{1} \alpha_{2}} \label{eq:SN:5.23}.
\end{align}

For $\sigma^{(j)}$ satisfying \eqref{eq:SN:5.21}, we now construct the sequence $c_{p}$ which decays rapidly as $p \rightarrow \infty$. The recursion relation \eqref{eq:SN:5.10}--\eqref{eq:SN:5.11} uniquely defines the values of $\eta_{p}$. After letting $c_{1} = \alpha_{1}$, we define
\begin{align}
c_{p} = \alpha_{p} \eta_{p} \eta_{p-1} \ldots \eta_{2} \label{eq:SN:5.24}
\end{align}
for all $p \geq 2$. This sequence satisfies \eqref{eq:SN:5.7}--\eqref{eq:SN:5.8} by construction. Furthermore, we observe that $\eta_{p} = - F_{p}(\sigma^{(j)})$. Repeating the arguments which gave \eqref{eq:SN:5.19} and \eqref{eq:SN:5.20} with $2$ replaced by $p$ gives
\begin{align}
\frac{-2}{\sigma_{p}^{(j)} \alpha_{p}} < \eta_{p} < \frac{-1}{\sigma_{p}^{(j)} \alpha_{p}}. \label{eq:SN:5.26}
\end{align}
Moreover, from \eqref{eq:SN:5.4} we have that $\alpha_{p} = \mathcal O(p^{4})$ as $p\rightarrow \infty$, and hence from \eqref{eq:SN:5.26} we obtain that $\eta_{p} = \mathcal O(p^{-4})$. Thus it follows from \eqref{eq:SN:5.24} that $c_{p} \rightarrow 0$ as $p\rightarrow \infty$, and this convergence is very fast, namely $\mathcal O(C^{p} (p-1)!^{-4})$ as $p \rightarrow \infty$, for some positive constant $C = C(\sigma^{(j)},\mu,\Omega,a,m,j)$. Therefore the solution $\theta^{(j)}(\boldsymbol x,t)$ given by \eqref{eq:SN:5.2} with $\sigma$ replaced by $\sigma^{(j)}$, and $c_{p}$ as defined by \eqref{eq:SN:5.24}, lies in any Sobolev space, it is $C^{\infty}$ smooth, and even real-analytic.

We substitute for $\alpha_{1}$ and $\alpha_{2}$ from \eqref{eq:SN:5.4} into the bound \eqref{eq:SN:5.23} to obtain estimates on $\sigma^{(j)}$ given by
\begin{align}
\frac{a \mu m j^{2}(j^{4}+j^{2})}{2^{3} \Omega^{2} m^{2} (j^{4}+j^{2} + 4 m^{2}) + 2 \mu^{2} j^{4} }  - \kappa (j^{4} + j^{2} + 4m^{2})^{\gamma} < \sigma^{(j)}  \label{eq:SN:5.27}
\end{align}
and
\begin{align}
\sigma^{(j)} < \frac{2 a \mu m j^{2}(j^{4}+j^{2})}{2^{3} \Omega^{2} m^{2} (j^{4}+j^{2} + m^{2}) + 2 \mu^{2} j^{4} } - \kappa (j^{4} + j^{2} + m^{2})^{\gamma} \label{eq:SN:5.28}.
\end{align}

We recall the role of the constants in the above expressions \eqref{eq:SN:5.27}--\eqref{eq:SN:5.28} for the bounds on $\sigma^{(j)}$.
The physical parameters are $\kappa$ the coefficient of thermal diffusivity, $\Omega$ the rotation rate of the system, $\mu$ is related to the underlying magnetic field, and $a$ is the magnitude of the steady {\em buoyancy} $\Theta_{0}$. Expressions  \eqref{eq:SN:5.27} and \eqref{eq:SN:5.28} indicate wide ranges of the physical parameters for which there exist eigenvalues $\sigma^{(j)}$ which are positive as postulated in the construction of the solution to \eqref{eq:SN:5.1}. We note that in the Earth's fluid core $\kappa$ is very small.
\end{proof}

\begin{lemma} \label{lemma:SN:5.3}
Let $\gamma \in (0,1/2)$. Fix $a,\mu,\Omega,\kappa>0$ and an integer $m\geq 1$. There exists an integer $j_{0} = j_{0}(a,m,\mu,\Omega,\kappa,\gamma)$ such that for all $j \geq j_{0}$ there exists a $C^{\infty}$ smooth initial datum $\theta^{(j)}(0,\boldsymbol x)$ with $\Vert \theta^{(j)}(0,\cdot) \Vert_{L^{2}}=1$ and a $C^{\infty}$ smooth solution $\theta^{(j)}(t,\boldsymbol x)$ of the initial value problem associated with the linearized $(MG)$\  equation \eqref{eq:linearequation}, such that
\begin{align}
\Vert \theta^{(j)}(t,\cdot) \Vert_{L^{2}} \geq \exp\left( j^{2} C_{*} t\right)
\end{align}
for all $t>0$, where $C_{*} = C_{*}(a,m,\mu,\Omega,\kappa,j_{0})$ is a positive constant defined in \eqref{eq:SN:5.29} below.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:SN:5.3}]
For any $j \geq 1$,  Lemma~\ref{lemma:SN:5.2} guarantees the existence of an eigenvalue $\sigma^{(j)}$ of the linear $(MG)$\  operator $L$, with associated $C^{\infty}$ smooth eigenfunction $\theta^{(j)}(0,\boldsymbol x)$ given by letting $t=0$ in \eqref{eq:SN:5.2}. Then $\theta^{(j)}(t,\boldsymbol x) = \exp(\sigma^{(j)} t ) \theta^{(j)}(0,\cdot)$ is a solution of \eqref{eq:linearequation}. It is clear that one may normalize $\theta^{(j)}(0,\cdot)$ to have
$L^{2}$ norm equal to $1$, and hence $\Vert \theta^{(j)}(t,\cdot) \Vert_{L^{2}} = \exp(\sigma^{(j)} t)$.

The lemma is then proven if we pick $j$ large enough so that $\sigma^{(j)} \geq j^{2} C_{*}$ for some positive constant $C_{*}$ (independent of $j$). This is guaranteed by the lower bound \eqref{eq:SN:5.27} for $\sigma^{(j)}$, if we pick $j \geq j_{0}$, where $j_{0}$ is a large enough fixed integer such that
\begin{align}
C_{*} = \frac{a \mu m (j_{0}^{4} + j_{0}^{2})}{2^{3} \Omega^{2} m^{2} (j_{0}^{4} + j_{0}^{2} + 4 m^{2}) + 2\mu^{2} j_{0}^{4}} - \frac{\kappa}{j_{0}^{2}} (j_{0}^{4}+j_{0}^{2} + 4m^{2})^{\gamma} > 0. \label{eq:SN:5.29}
\end{align}
  Note that when $\gamma \in(0,1/2)$ such a $j_{0}$ always exists, independently of the size of the physical parameters.
\end{proof}

We now have all necessary ingredients to conclude the proof of Theorem~\ref{thm:SN:5.1}.
\begin{proof}[Proof of Theorem~\ref{thm:SN:5.1}]
Let $T>0$ and $K>0$ be arbitrary, and let $Y \subset L^{2}$ be a Sobolev space. Pick an integer $m\geq 1$, and an amplitude $a>0$. For these fixed $a,m$ and physical parameters $\mu,\Omega,\kappa,\gamma>0$, define $j_0$ and $C_*$ as in Lemma~\ref{lemma:SN:5.3}.
For any $j \geq j_0$, Lemma~\ref{lemma:SN:5.3} guarantees that there exists a $C^\infty$ smooth initial condition $\theta^{(j)}(0,\boldsymbol x)$ which we re-normalize to have $\Vert \theta^{(j)}(0,\cdot) \Vert_{Y} =1$, such that the associated solution $\theta^{(j)}(t,\boldsymbol x)$ of the Cauchy problem \eqref{eq:linearequation}--\eqref{eq:steadystatedef} satisfies
\begin{align}
\Vert \theta^{(j)}(t,\cdot) \Vert_{L^2} \geq \exp( j^2 C_* t) \Vert \theta^{(j)}(0,\cdot) \Vert_{L^2} \label{eq:SN:thm:lowerbound}
\end{align}
for all $t>0$. We note that this solution to the linear equation is the unique\footnote{The proof of uniqueness of finite energy solutions to the linear equation follows from a representation of the solution as a Fourier series (see \cite{FriedlanderVicol11c}, Proposition~2.8).} solution in $L^\infty(0,T;L^2)$. We now claim that there exists a sufficiently large $j_* \geq j_0$ such that
\begin{align}
\Vert \theta^{(j_*)}(T/2,\cdot) \Vert_{L^2} \geq 2  K \label{eq:SN:thm:linear}
\end{align}
which would then conclude the proof of the Theorem, since $\Vert \theta^{(j_*)}(0,\cdot) \Vert_{Y} =1$. After a short calculation it is clear that \eqref{eq:SN:thm:linear} follows from \eqref{eq:SN:thm:lowerbound} if we manage to prove that
\begin{align}
 \exp\left( \frac{j_*^2 C_* T}{2}\right)  \Vert \theta^{(j_*)}(0,\cdot) \Vert_{L^2} \geq 2 K. \label{eq:SN:thm:suff:cond}
\end{align}
But  from the definition of $\theta^{(j_*)}$ cf.~\eqref{eq:SN:5.2} above, we see that $\Vert \theta^{(j_*)}(0,\cdot) \Vert_{L^2} \geq c_1 = \alpha_1 \geq 1/ (C j_*^2)$, for some sufficiently large constant $C$, cf.~\eqref{eq:SN:5.4} above. The fact that for every fixed $T>0$ and $C_*>0$ the sequence $\exp(j^2 C_* T/2)/ (C j^2)$ diverges as $j\rightarrow \infty$, shows that there exits some sufficiently large $j_*$ such that \eqref{eq:SN:thm:suff:cond} holds, thereby concluding the proof of the theorem.
\end{proof}

\subsection{Non-linear ill-posedness in Sobolev spaces} \label{eq:sec:non-linear:ill}
Having established that the {\em linearized} $(MG)$\  are ill-posed in $L^2$, we now turn to address the Hadamard ill-posedness of the {\em full nonlinear} $(MG)$\  equations. The ill-posedness of non-linear partial differential equations may have different sources, of varying degree of gravity: finite time blow-up, non-uniqueness, or non-smoothness of the solution map, to name a few. As in \cite{FriedlanderVicol11c} for the case $\kappa=0$, here we show that the fractionally diffusive $(MG)$\  equations, with $0 <\gamma < 1/2$ do not posses a Lipschitz continuous solution map.
Recall the definition of Lipschitz well-posedness for the nonlinear equation (see~\cite[Definition 1.1]{GuoNguyen10}, or \cite[Definition 2.9]{FriedlanderVicol11c}):

\begin{definition}[\bf Lipschitz local well-posedness] \label{def:well}
Let $Y \subset X \subset W^{1,4}$ be Banach spaces. The Cauchy problem for the $(MG)$\  equation
\begin{align}
&\partial_t \Theta + \boldsymbol U \cdot \nabla \Theta + \kappa (-\Delta)^\gamma \Theta = S  \label{eq:nonlinear:ill:1}\\
&\nabla \cdot \boldsymbol U =0, \  U_{j} =  M_{j} \Theta \label{eq:nonlinear:ill:2}
\end{align}is locally Lipschitz $(X,Y)$ {\em well-posed}, if there exist continuous functions $T,K: [0,\infty)^{2} \rightarrow (0,\infty)$,  the time of existence and the Lipschitz constant, so that for every pair of initial data $\Theta^{(1)}(0,\cdot), \Theta^{(2)}(0,\cdot) \in Y$ there exist unique solutions $\Theta^{(1)}, \Theta^{(2)} \in L^{\infty}(0,T;X)$ of the initial value problem associated to  \eqref{eq:nonlinear:ill:1}--\eqref{eq:nonlinear:ill:2}, that additionally satisfy
\begin{align}
\Vert \Theta^{(1)}(t,\cdot) -\Theta^{(2)}(t,\cdot) \Vert_{X} \leq K \Vert \Theta^{(1)}(0,\cdot) - \Theta^{(2)}(0,\cdot) \Vert_{Y}\label{eq:def:well}
\end{align}
for every $t\in [0,T]$, where $T = T(\Vert \Theta^{(1)}(0,\cdot) \Vert_{Y}, \Vert \Theta^{(2)}(0,\cdot)\Vert_{Y})$ and $K  = K (\Vert \Theta^{(1)}(0,\cdot)\Vert_{Y},\Vert \Theta^{(2)}(0,\cdot)\Vert_{Y})$.
\end{definition}
\begin{remark}
 Clearly the time of existence $T$ and the Lipschitz constant $K$ also depend on $\Vert S \Vert_{L^\infty(0,\infty;Y)}$, and on $\kappa$, but we have omitted this dependence in Definition~\ref{def:well} since it is the same for both solutions $\Theta^{(1)}$ and $\Theta^{(2)}$.
\end{remark}
\begin{remark}
 If $\Theta^{(2)}(t,\cdot)=0$ and $X=Y$, Definition~\ref{def:well} recovers the usual definition of local well-posedness with a continuous solution map. However, Defintion~\ref{def:well} allows the solution map to lose regularity, which is usually needed in order to obtain Lipschitz continuity of the solution map for first order quasi-linear equations. Hence, the typical pairs of spaces $(X,Y)$ that we have in mind here are $X = H^{s}$, and  $Y=H^{s+1}$, with $s>1 + 3/4$.
 \end{remark}

The main result of this section is the following theorem.

\begin{theorem}[\bf Nonlinear ill-posedness in Sobolev spaces] \label{thm:nonlinear:ill}
The $(MG)$\  equations \eqref{eq:nonlinear:ill:1}--\eqref{eq:nonlinear:ill:2}, with $\gamma \in (0,1/2)$ are locally Lipschitz $(X,Y)$ {\em ill-posed} in Sobolev spaces $Y\subset X$ embedded in $W^{1,4}({\mathbb T}^3)$, in the sense of Definition~\ref{def:well}.
\end{theorem}

 For the purpose of our ill-posedness result, we shall let $\Theta^{(2)}(t,\boldsymbol x)$ be the steady state $\Theta(x_{3})$ introduced earlier in \eqref{eq:steadystatedef}. We consider $X$ to be a Sobolev space with high enough regularity so that $\partial_{t} \Theta \in L^{\infty}(0,T;L^{2})$, which implies that $\Theta$ is weakly continuous on $[0,T]$ with values in $X$, making sense of the initial value problem associated to \eqref{eq:nonlinear:ill:1}--\eqref{eq:nonlinear:ill:2}. The proof of Theorem~\ref{thm:nonlinear:ill} follows from the strong {\em linear} ill-posedness obtained in Theorem~\ref{thm:SN:5.1} and a fairly generic perturbative argument (cf.~\cite[Thorem 2]{Renardy09} or~\cite[pp.~183]{Tao06}). The proof of Theorem~\ref{thm:nonlinear:ill} follows the lines of the proof for the non diffusive problem given in \cite{FriedlanderVicol11c}.

\begin{proof}[Proof of Theorem~\ref{thm:nonlinear:ill}]
Since the Sobolev space $X$ embeds in $H^{1}$, and $\gamma \in (0,1/2)$, the linearized operator $L \Theta = - M_{3}\Theta\, \partial_{3} \Theta_{0} - \kappa (-\Delta)^\gamma \Theta$ maps $X$ continuously into $L^{2}$, and since $X \subset W^{1,4}$, the nonlinearity $N\Theta = - M_{j}\Theta\, \partial_{j} \Theta$ is bounded as  $\Vert N \Theta \Vert_{L^{2}} \leq \Vert \nabla \Theta \Vert_{L^{4}}^{2} \leq C \Vert \Theta \Vert_{X}^{2}$, for some  constant $C>0$.

Fix the steady state $\Theta_{0}(x_{3}) \in Y$, as given by \eqref{eq:steadystatedef}. Also, fix a smooth function $\psi_{0} \in Y$, normalized to have $\Vert \psi_{0} \Vert_{Y} = 1$, to be chosen precisely later. The proof is by contradiction. Assume that the Cauchy problem for the  $(MG)$\  equation \eqref{eq:nonlinear:ill:1}--\eqref{eq:nonlinear:ill:2} is Lipschitz locally well-posed in $(X,Y)$. Consider $\Theta^{(2)}(0,\boldsymbol x) = \Theta_{0}(x_{3})$, so that $\Theta^{(2)}(t,\boldsymbol x) = \Theta_{0}(x_{3})$ for any $t>0$. Also let
\begin{align*}
\Theta^{\epsilon}(0,\boldsymbol x) = \Theta_{0}(x_{3})  + \epsilon \psi_{0}(\boldsymbol x),
\end{align*} for every $0 < \epsilon < \Vert \Theta_{0} \Vert_{Y}$. To simplify notation we write $\Theta^{\epsilon}$ instead of $\Theta^{(1,\epsilon)}$. By Definition~\ref{def:well}, for every $\epsilon$ as before there exists a positive time $T = T (\Vert \Theta_{0}\Vert_{Y},\Vert \Theta^{\epsilon}\Vert_{Y})$ and a positive Lipschitz constant $ K = (\Vert \Theta_{0}\Vert_{Y},\Vert \Theta^{\epsilon}\Vert_{Y})$ such that by \eqref{eq:def:well} and the choice of $\psi_{0}$ we have
\begin{align}
\Vert \Theta^{\epsilon}(t,\cdot) - \Theta_{0}(\cdot) \Vert_{X} \leq K \epsilon \label{eq:Lip:bound}
\end{align}for all $t\in [0,T]$. We note that since $\Vert \Theta^{\epsilon}(0,\cdot) \Vert_{Y} \leq \Vert \Theta_{0} \Vert_{Y} + \epsilon \leq 2 \Vert \Theta_{0} \Vert_{Y}$, due to the continuity of $T$ and $K$ with respect to the second coordinate, we may choose $K = K(\Vert \Theta_{0} \Vert_{Y}) >0$ and $T =T (\Vert \Theta_{0} \Vert_{Y})>0$ independent of $\epsilon \in (0,\Vert \Theta_{0} \Vert_{Y})$, such that  \eqref{eq:Lip:bound} holds on $[0,T]$.

Writing $\Theta^{\epsilon}$ as an $\mathcal O(\varepsilon)$  perturbation of $\Theta_0$, we define
\begin{align}
\psi^{\epsilon}(t,\boldsymbol x) = \frac{\Theta^{\epsilon}(t,\boldsymbol x) - \Theta_0(x_3)}{\varepsilon},
\end{align}
for all $t \in [0,T]$ and all $\epsilon$ as before.  It follows from \eqref{eq:Lip:bound} that $ \psi^\epsilon$   is uniformly bounded with respect to $\epsilon$ in $L^{\infty}(0,T;X)$. Therefore, there exists a function $\psi$, the weak-$*$ limit of $\psi^{\epsilon}$ in $L^\infty(0,T;X)$. Note that $\psi^{\epsilon}$ solves the Cauchy problem
  \begin{align}
    \partial_t \psi^\epsilon &= L \psi^\epsilon + {\epsilon} N(\psi^\epsilon) \label{eq:psieps}\\
    &\psi^{\epsilon}(0,\cdot) = \psi_{0}.
  \end{align}
  Due to the choice of $X$, we have the bound
  \begin{align}
  \Vert N \psi^{\epsilon} \Vert_{L^{2}} \leq C \Vert \psi^{\epsilon} \Vert_{X}^{2} \leq C K^{2} \label{eq:Nbound},
  \end{align}and from \eqref{eq:psieps} we obtain that $\partial_{t} \psi^{\epsilon}$ is uniformly bounded with respect to $\epsilon$ in $L^{\infty}(0,T;L^{2})$. Therefore the convergence $\psi^{\epsilon} \rightarrow \psi$ is strong when measured in $L^{2}$. Sending $\epsilon$ to $0$ in \eqref{eq:psieps}, and using \eqref{eq:Nbound}, it follows that
  \begin{align*}
    \partial_t \psi &= L \psi\\
    \psi(0,\cdot)&= \psi_{0}
  \end{align*}
  holds in $L^\infty(0,T;L^2)$, and this solution is unique since the problem is now linear. In addition, the solution $\psi$ inherits from \eqref{eq:Lip:bound} the upper bound
  \begin{align}
  \Vert \psi(t,\cdot) \Vert_{L^{2}} \leq K \label{eq:Lip:2}
  \end{align} for all $t\in[0,T]$. But this is a contradiction with Theorem~\ref{thm:SN:5.1}. Due to the existence of eigenfunctions for the linearized operator with arbitrarily large eigenvalues, one may choose $\psi_{0}$ (as in Lemma~\ref{lemma:SN:5.3}) to yield a large enough eigenvalue so that in time $T/2$ the solution grows to have $L^{2}$ norm larger than $2K$ (similarly to the proof of Theorem~\ref{thm:SN:5.1}), therefore contradicting \eqref{eq:Lip:2}.
\end{proof}

\section{The regime $\gamma = 1/2$: a dichotomy in terms of the size of the data} \label{sec:=1/2}

If the initial data is small with respect to $\kappa$, then one may use energy estimates to show that there exists a unique global smooth solution evolving from this data (cf.~Section~\ref{sec:=1/2_small_data} below). However, the proof does not apply for the case of large data, not even to prove the local existence of solutions. In the case of large data, we may construct a steady state such that solutions evolving from initial data which is arbitrarily close to this steady state diverge from it at an arbitrarily large rate, for positive time (cf.~Section~\ref{sec:=1/2_large_data} below), i.e. the equations are Hadamard ill-posed.

\subsection{Ill-posedness for $\gamma=1/2$ and large data} \label{sec:=1/2_large_data}
In Section~\ref{sec:<1/2} above we have shown that for $\gamma<1/2$ the $(MG)$\  equations are Hadamard ill-posed in Sobolev spaces, in the sense that there is no Lipschitz continuous solution map (see Theorem~\ref{thm:nonlinear:ill}). The main ingredient in the proof of ill-posedness for $\gamma \in (0,1/2)$ was to show that the linearized $(MG)$\  operator
\begin{align}
L \theta = - M_3 \theta \partial_3 \Theta_0 - \kappa (-\Delta)^\gamma \theta
\end{align}
where
\begin{align}
\Theta_0 = a \sin(m x_3)
\end{align}
is a steady state associated to the source term $S =\kappa a m^{2\gamma} \sin(m x_3)$, has a sequence of eigenvalues $\sigma^{(j)}$ which diverge to infinity as $j \rightarrow \infty$ (see Lemmas~\ref{lemma:SN:5.2} and \ref{lemma:SN:5.3}).

We emphasize that it is {\em only} in the proof of Lemma~\ref{lemma:SN:5.3} where $\gamma < 1/2$, rather than $\gamma \leq 1/2$ is used. Indeed, in order to prove that given {\em any} real $a>0$ and any integer $m \geq 1$ there exists some sufficiently large integer $j_0$ such that the constant $C_*$ of \eqref{eq:SN:5.29} is strictly positive, $\gamma < 1/2$ is both necessary and sufficient. In the case $\gamma=1/2$, we can prove the following alternative to Lemma~\ref{lemma:SN:5.3}, which states that only if $a$ is sufficiently large with respect to $\kappa$ (in terms of $\mu$ and $\Omega$), then one obtains a sequence of eigenvalues which diverge to $\infty$. Namely:

\begin{lemma} \label{lemma:SN:5.4}
Let $\gamma = 1/2$. Fix $a,\mu,\Omega$, and $m$. For values of $a$ such that
\begin{align}
\kappa < \frac{a \mu m}{2^5 \Omega^2 m^2 + 2 \mu^2} \label{eq:SN:cond:a}
\end{align}
and all integers $j \geq m$, the statement of Lemma~\ref{lemma:SN:5.3} holds with the constant $C_*$ given by
\begin{align}
C_*  = \frac{a \mu m}{2^5 \Omega^2 m^2 + 2 \mu^2} - \kappa \label{eq:SN:5.30}.
\end{align}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:SN:5.4}]
 The proof follows from the lower bound \eqref{eq:SN:5.27} on the eigenvalue $\sigma^{(j)}$, similarly to the proof of Lemma~\ref{lemma:SN:5.3}. The role of condition \eqref{eq:SN:cond:a} is to guarantee that there exists a large enough $j_0$ such that
\begin{align*}
 \frac{a \mu m (j_{0}^{4} + j_{0}^{2})}{2^{3} \Omega^{2} m^{2} (j_{0}^{4} + j_{0}^{2} + 4 m^{2}) + 2\mu^{2} j_{0}^{4}} - \frac{\kappa}{j_{0}^{2}} (j_{0}^{4}+j_{0}^{2} + 4m^{2})^{\gamma} > 0.
\end{align*}
To avoid repetition we omit further details.
\end{proof}
For those values of $a$ for which \eqref{eq:SN:cond:a} holds, Lemma~\ref{lemma:SN:5.4} shows that the operator $L$ has unbounded spectrum in the unstable region, and hence one may prove with virtually {\em no modifications} from the $\gamma \in (0,1/2)$ case, that the equations are ill-posed. More precisely we have
\begin{theorem}[\bf Ill-posedness for large data] \label{thm:ill:1/2}
 Let $Y \subset X \subset W^{1,4}({\mathbb T}^3)$ be Sobolev spaces, and let $\gamma=1/2$. Given $\kappa, \mu, \Omega >0$, fix an integer $m\geq 1$. Let $a>0$ be sufficiently large such that \eqref{eq:SN:cond:a} holds and let $\Theta_0 ^{(1)}= a \sin(m x_3) \in Y$ be a steady state of \eqref{eq:1}. Then, given any $T>0$ and any $K>0$, there exists an initial condition $\Theta_0^{(2)} \in Y$ and a corresponding solution $\Theta^{(2)} \in L^\infty(0,T;X)$ of the Cauchy problem \eqref{eq:1}--\eqref{eq:3}, such that
 \begin{align}
\sup_{t\in [0,T]} \Vert  \Theta^{(2)}(t,\cdot) - \Theta^{(1)}(t,\cdot) \Vert_{X} \geq 2 K \Vert  \Theta^{(2)}_0 - \Theta^{(1)}_0 \Vert_{Y}
\end{align}
where $ \Theta^{(1)}(t,\cdot) = \Theta^{(1)}_0$. That is, the $(MG)$\  equations are locally Lipschitz (X,Y) ill-posed when the data is large.
\end{theorem}
The proof of Theorem~\ref{thm:ill:1/2} is the same as the proof of Theorem~\ref{thm:nonlinear:ill} above. The only difference in the case $\gamma=1/2$ is to use Lemma \ref{lemma:SN:5.4} to show that the linear equations have unbounded unstable spectrum. We omit  details.

\subsection{Well-posedness for $\gamma=1/2$ and small data} \label{sec:=1/2_small_data}

As shown in the previous section, we can exhibit initial data, for which the system (\ref{eq:1})--(\ref{eq:2}) is ill-posed in the above described sense. Note also, that the proof of Theorem \ref{thm:121:local}, fails for the value $\gamma = 1/2$. Thus, $\gamma=1/2$ indeed is the limit of the local well-posedness theory. Nonetheless, we still can prove that the considered system is globally well-posed for small data.

\begin{theorem}[{\bf Global existence for small data}] \label{thm:gamma=1/2:small}
	Let $s>5/2$ and assume that the initial data $\Theta_0 \in H^s({\mathbb T}^3)$ and $S \in L^2(0,\infty;H^{s-\gamma}({\mathbb T}^3))$ have zero-mean on ${\mathbb T}^{3}$. There exists a sufficiently small constant $\varepsilon>0$ depending on $\kappa$, such that if $\Vert \Theta_0\Vert_{L^2}^\alpha\Vert \Theta_{0} \Vert_{H^s}^{1-\alpha} + \Vert S \Vert_{L^\infty(0,\infty;H^{s-\gamma})} \leq \varepsilon$, where $\alpha=1-(5/2)/s$, then the unique smooth solution
	\begin{align}
		\Theta \in L^\infty(0,T;H^s({\mathbb T}^3)
	\end{align}
	 of the Cauchy problem \eqref{eq:1}--\eqref{eq:M:3} is global in time.
\end{theorem}
\begin{proof}
	We proceed as in the proof of Theorem \ref{thm:121:global} and obtain the energy estimate
	\begin{align}\label{eq:3.1}
		&\frac{1}{2}\frac{d}{dt}\|\Lambda^s\Theta\|^2_{L^2} + \kappa\|\Lambda^{s+\frac{1}{2}}\Theta\|^2_{L^2} \leq \|\Lambda^{s-\frac{1}{2}}S\|_{L^2}\|\Lambda^{s+\frac{1}{2}}\Theta\|_{L^2} + \left| \int_{{\mathbb R}^3} \Lambda^{s-\frac{1}{2}}(\boldsymbol U\cdot \nabla \Theta)\Lambda^{s+\frac{1}{2}}\Theta \right|.
	\end{align}
The second term on the right side is estimated using the product estimate in Proposition \ref{prop:calculus}. Thus we obtain, similarly to (\ref{eq:2.8})
\begin{align}
	&\|\Lambda^{s-\frac{1}{2}}(\boldsymbol U \cdot \nabla \Theta)\|_{L^2} \leq C(\|\nabla \Theta\|_{L^\infty} + \|\boldsymbol U\|_{L^\infty})\|\Lambda^{s+\frac{1}{2}}\Theta\|_{L^2}.
\end{align}
Since $s > 5/2$,  we get
\begin{align}\label{eq:3.3}
	& \|\boldsymbol U\|_{L^\infty} + \|\nabla \Theta\|_{L^\infty} \leq C\|\Theta\|_{L^2}^\alpha\|\Lambda^s \Theta\|_{L^2}^{1-\alpha},
\end{align}
where $\alpha = 1-(n/2+1)/s$.
Combining estimates (\ref{eq:3.1})-(\ref{eq:3.3}) gives
\begin{align}
	&\frac{1}{2}\frac{d}{dt}\|\Lambda^s\Theta\|^2_{L^2} + \kappa\|\Lambda^{s+\frac{1}{2}}\Theta\|^2_{L^2} \leq \frac{1}{2\kappa}\|\Lambda^{s-\frac{1}{2}}S\|^2_{L^2} + \frac{\kappa}{2}\|\Lambda^{s+\frac{1}{2}}\Theta\|^2_{L^2} + C\|\Theta\|_{L^2}^\alpha\|\Lambda^s \Theta\|_{L^2}^{1-\alpha}\|\Lambda^{s+\frac{1}{2}} \Theta\|_{L^2}^2,
\end{align}
which in turn leads to
\begin{align}\label{eq:3.5}
	&\frac{1}{2}\frac{d}{dt}\|\Lambda^s\Theta\|^2_{L^2} + \frac{\kappa}{2}\|\Lambda^{s+\frac{1}{2}}\Theta\|^2_{L^2} \leq \frac{1}{2\kappa}\|\Lambda^{s-\frac{1}{2}}S\|^2_{L^2} +  C\|\Theta\|_{L^2}^\alpha\|\Lambda^s \Theta\|_{L^2}^{1-\alpha}\|\Lambda^{s+\frac{1}{2}} \Theta\|_{L^2}^2.
\end{align}
We obtain the desired result as in the proof of Theorem \ref{thm:121:global}.
\end{proof}
\begin{remark}
	Note that conditions in Theorem \ref{thm:121:global} are consistent with the above theorem if we set $\gamma=1/2$.
\end{remark}

\begin{remark}
 We note that the ill-posedness result in the case $\gamma=1/2$ requires the constant defined in \eqref{eq:SN:5.30} to be positive. This does not hold when the value of $a$ is sufficiently small (depending on $\kappa, \mu$, and $\mu$) no matter what value of $m\geq 1$ is picked. Recalling that $a$ measures the magnitude of the initial data associated with $\Vert \Theta_0 \Vert_{L^2}$, we observe that the well-posedness result  when $\gamma = 1/2$ is only proven in the case of small data and small force (see Theorem~\ref{thm:gamma=1/2:small}), and hence for $a$ sufficiently small. Therefore our large data ill-posedness result is consistent with the small-data well-posedness result when $\gamma = 1/2$.
\end{remark}

\section{Improvement in regularity for ``well-prepared'' data and source} \label{sec:well-prepared}

In this section we explore the following observation:  if the frequency support of $\Theta$ lies on a suitable plane, then the operator $\boldsymbol M$ is ``mild'' when it acts on $\Theta$, i.e. it behaves like an order $0$ operator, and hence the corresponding velocity $\boldsymbol U$ is as smooth as $\Theta$. This enables us to obtain improved well-posedness results over the generic setting when no conditions on the Fourier spectrum of the initial data (and source term) are imposed. For instance the local existence an uniqueness of smooth solutions holds even if $0<\gamma<1/2$, a setting in which we know that for generic initial data the problem is ill-posed.

To be more precise let $q \in {\mathbb Q}$ be a rational number\footnote{Since we are in the periodic setting when the frequency is supported on ${\mathbb Z}^{3}$.} with $q \neq 0$. We define the frequency plane $\mathcal P_{q}$ to be the set
\begin{align}
\mathcal P_{q} = \{ \boldsymbol k = (k_{1},k_{2},k_{3}) \in {\mathbb Z}^{3} \colon k_{2} = q k_{1} \}. \label{eq:Pq:def}
\end{align}
We shall need the following straightforward observation:
\begin{proposition}
\label{prop:support}
Assume $f,g$ are smooth ${\mathbb T}^{3}$ periodic functions, with frequency support $\mathop {\rm supp} \nolimits(\hat f), \mathop {\rm supp} \nolimits(\hat g) \subset \mathcal P_{q}$ for some $q \in {\mathbb Q}\setminus\{0\}$. Then we have $\mathop {\rm supp} \nolimits(\hat{f\pm g}), \mathop {\rm supp} \nolimits( \hat{fg} ), \mathop {\rm supp} \nolimits (\hat {M_{j} f}) \subset \mathcal P_{q}$, for all $j \in \{1,2,3\}$.
\end{proposition}
\begin{proof}[Proof of Propostion~\ref{prop:support}]
Clearly
$\mathop {\rm supp} \nolimits(\hat{f + g}) = \mathop {\rm supp} \nolimits (\hat f + \hat g) \subset \mathop {\rm supp} \nolimits (\hat f) \cup \mathop {\rm supp} \nolimits (\hat g) \subset  \mathcal P_{q}$.
To see that that the frequency support of the function $M_{j} f$ lies on  $\mathcal P_{q}$, we just note that $\mathop {\rm supp} \nolimits(\hat{M_{j} g}) = \mathop {\rm supp} \nolimits( \hat{M_{j}} \hat{f})  \subset \mathop {\rm supp} \nolimits(\hat{M_{j}}) \cap \mathop {\rm supp} \nolimits(\hat f) \subset \mathcal P_{q}$. To analyze the frequency support of $\hat{fg} = \hat f \ast \hat g$, note that
$\mathop {\rm supp} \nolimits( \hat f \ast \hat g) \subset \mathop {\rm supp} \nolimits(\hat f ) + \mathop {\rm supp} \nolimits(\hat g) \subset \mathcal P_{q}$, since $\mathcal P_{q}$ is closed under addition of vectors, concluding the proof.
\end{proof}

\begin{lemma}\label{lemma:M:order:0}
If $f$ is a smooth ${\mathbb T}^{3}$ periodic function with $\hat f(k_{1},k_{2},0) = 0$ for all $k_{1},k_{2} \in {\mathbb Z}$, and such that  $\mathop {\rm supp} \nolimits(\hat f) \subset \mathcal P_{q}$ for some $q \in {\mathbb Q}\setminus\{0\}$, then there exists a universal constant $C_{\ast} = C_{\ast} (q,\Omega,\beta,\eta)>0$ such that
\begin{align}
\left| (\hat{M_{j} f})(\boldsymbol k)\right| = \left| \hat{M_{j}}(\boldsymbol k) \hat{f}(\boldsymbol k)\right| \leq C_{\ast}|\hat{f}(\boldsymbol k)| \label{eq:M:order:0}
\end{align}
for all $\boldsymbol k \in {\mathbb Z}^{3}$, and for all $j \in \{1,2,3\}$. Additionally, the constant $C_*$ blows-up as $q \rightarrow 0$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:M:order:0}] It is clear that \eqref{eq:M:order:0} has to be proven only for $\boldsymbol k$ such that $k_{3} \neq 0$, since otherwise we have that $\hat{f}(\boldsymbol k) = 0$ and the statement holds trivially. We now consider each of the cases $j \in \{1,2,3\}$. Without loss of generality take $q>0$.

For $j=1$, a short algebraic computations gives
\begin{align}
\left |\hat{M}_{1} (\boldsymbol k) \right| &\leq \left( 2\Omega (2 q + q^{3})  + q^{2} \beta^{2}/\eta \right) \frac{k_{1}^{3} k_{3} + k_{1} k_{3}^{3}}{2\Omega^{2} k_{3}^{4}+ q^{4} \beta^{4}/\eta^{2} k_{1}^{4}} \label{eq:M1:estimate:order:0}
\end{align}
from which it follows that $|\hat{M}_{1}(\boldsymbol k)| \leq C$, for a suitable constant $C$, by using the inequality $k_{1}^{3} k_{3} + k_{1} k_{3}^{3} \leq k_{1}^{4}  + k_{3}^{4}$. Similarly to \eqref{eq:M1:estimate:order:0} it follows that for $\boldsymbol k \in \mathcal P_{q}$ we have
\begin{align}
\left| \hat{M}_{2} (\boldsymbol k)\right| \leq \left( 2\Omega (2+q^{2})+ q^{3} \beta^{2}/\eta \right) \frac{k_{1}^{3} k_{3} + k_{1} k_{3}^{3}}{2\Omega^{2} k_{3}^{4}+ q^{4} \beta^{4}/\eta^{2} k_{1}^{4}}
\end{align}
and
\begin{align}
\left| \hat{M}_{3} (\boldsymbol k)\right| \leq  \left( q^{2} (1+q^{2}) \beta^{2}/\eta\right) \frac{k_{1}^{4} }{2\Omega^{2} k_{3}^{4}+ q^{4} \beta^{4}/\eta^{2} k_{1}^{4}}
\end{align}
which concludes the proof of the lemma upon using $k_{1}^{3} k_{3} + k_{1} k_{3}^{3} \leq k_{1}^{4}  + k_{3}^{4}$.
\end{proof}

\subsection{Local existence and uniqueness for $0 < \gamma < 1$}\label{sec:well-prepared:local}
The main result of this section is:

\begin{theorem}[{\bf Local existence}] \label{thm:well-prepared:local}
Let $\gamma \in (0,1)$, and fix $s>5/2-\gamma$. Assume that $\Theta_0 \in H^s({\mathbb T}^3)$ and $S \in L^\infty(0,\infty;H^{s-\gamma}({\mathbb T}^3))$ have zero-mean on ${\mathbb T}^{3}$ and are such that
\begin{align}
\mathop {\rm supp} \nolimits(\hat\Theta_{0}) \subset \mathcal P_{q} \mbox{ and } \mathop {\rm supp} \nolimits(\hat{S}(t,\cdot)) \subset  \mathcal P_{q},
\end{align} for some fixed $q \in {\mathbb Q}\setminus\{0\}$ and all $t\geq 0$. Then there exists a $T>0$ and a unique smooth solution
\begin{align} \Theta \in L^\infty(0,T;H^s({\mathbb T}^3)) \cap L^2(0,T;H^{s+\gamma}({\mathbb T}^3))
\end{align} of the Cauchy problem  \eqref{eq:FractionalMG:1}--\eqref{eq:FractionalMG:2}, such that
\begin{align}
  \mathop {\rm supp} \nolimits(\hat\Theta (t,\cdot)) \subset \mathcal P_{q}
\end{align}
for all $t\in[0,T)$.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:well-prepared:local}]
In order to construct the local in time solution $\Theta$, with frequency support on $\mathcal P_q$, consider the sequence of approximations $\{ \Theta_n \}_{n\geq 1}$ given by the solutions of
\begin{align}
  &\partial_t \Theta_1 + (-\Delta)^\gamma \Theta_1 = S  \label{eq:Th:1:1}\\
  &\Theta_1(0,\cdot) = \Theta_0 \label{eq:Th:1:2}
\end{align}
and respectively
\begin{align}
  &\partial_t \Theta_n + \boldsymbol U_{n-1} \cdot \nabla \Theta_n + (-\Delta)^\gamma \Theta_n  = S \label{eq:Th:n:1}\\
  & \boldsymbol U_{n-1} = \boldsymbol M \Theta_{n-1} \label{eq:Th:n:2}\\
  &\Theta_n(0,\cdot)= \Theta_0 \label{eq:Th:n:3}
\end{align}
for all $n \geq 2$.
One may solve \eqref{eq:Th:1:1}--\eqref{eq:Th:1:2} explicitly using Duhamel's formula, and noting that the operator $\exp( (-\Delta)^\gamma t)$ is a Fourier multiplier with non-zero symbol given by $\exp(|\boldsymbol k|^\gamma t)$, it follows from Proposition~\ref{prop:support} that $\mathop {\rm supp} \nolimits(\hat{\Theta_1}(t,\cdot)) \subset \mathcal P_q$ for all $t\geq 0$. In addition, we have the bound
\begin{align}
\Vert \Lambda^s \Theta_1 \Vert_{L^\infty(0,T;L^2)}^2 + \Vert \Lambda^{s+\gamma} \Theta_1 \Vert_{L^2(0,T;L^2)}^2 \leq \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2 + T \Vert S \Vert_{L^\infty(0,T;H^{s-\gamma})}^2 \label{eq:Th:1:bound}
\end{align} for all $T>0$.
On the other hand, in order to solve \eqref{eq:Th:n:1}--\eqref{eq:Th:n:3} we appeal to Theorem~\ref{thm:app:linear}. Indeed, by the inductive assumption we have that $\Theta_{n-1} \in L^\infty(0,T;H^s)$  and also $\mathop {\rm supp} \nolimits(\hat{\Theta_{n-1}}(T,\cdot) ) \subset \mathcal P_q$ for all $T>0$. Hence, by applying Lemma~\ref{lemma:M:order:0} we have $\boldsymbol U_{n-1} \in L^\infty(0,T;H^s)$ and by Proposition~\ref{prop:support} we have $\mathop {\rm supp} \nolimits(\hat{\boldsymbol U_{n-1}}(t,\cdot))$ for all $t>0$. Therefore all the conditions of Theorem~\ref{thm:app:linear} are satisfied, by letting $v = \boldsymbol U_{n-1}$, and there hence exists a unique solution $\Theta_n \in L^\infty(0,T;H^s) \cap L^2(0,T;H^{s+\gamma})$ of \eqref{eq:Th:n:1}--\eqref{eq:Th:n:3}, such that $\mathop {\rm supp} \nolimits(\hat{\Theta_n}(t,\cdot)) \subset \mathcal P_q$ for all $t\in [0,T)$.

To prove that the sequence $\{\Theta_n\}$ converges, we first prove that it is bounded. Fix a time $T$ to be chosen later such that $T < \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2 / \Vert S \Vert_{L^\infty(0,T;H^{s-\gamma})}^2$. It hence follows from \eqref{eq:Th:1:bound} that
\begin{align}
  \Vert \Lambda^s \Theta_j \Vert_{L^\infty(0,T;L^2)}^2 + \Vert \Lambda^{s+\gamma} \Theta_j \Vert_{L^2(0,T;L^2)}^2 \leq 2 \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2 \label{eq:Th:inductive:bound}
\end{align}
holds when $j=1$. We assume inductively that \eqref{eq:Th:inductive:bound} holds for all $1 \leq j \leq n-1$, and proceed to prove that it holds for $j=n$. Since $\boldsymbol U_{n-1}$ is obtained from $\Theta_{n-1}$ by a bounded Fourier multiplier (cf.~Lemma~\ref{lemma:M:order:0}), there exists a positive constant $C_q$, depending on $q$ (and on the physical parameters $\Omega, \beta, \eta$), such that
\begin{align}
  \Vert \Lambda^s \boldsymbol U_{n-1}(t,\cdot) \Vert_{L^2}^2 \leq C_q  \Vert \Lambda^s \Theta_{n-1}(t,\cdot) \Vert_{L^2}^2 \label{eq:U:n-1:bound}
\end{align}
for all $t\geq 0$.
Applying $\Lambda^s$ to \eqref{eq:Th:n:1} and taking an $L^2$ inner product with $\Lambda^s \Theta_n$ we hence obtain
\begin{align}
& \frac{1}{2} \frac{d}{dt} \Vert \Lambda^{s} \Theta_n \Vert_{L^{2}}^{2} + \Vert \Lambda^{s+\gamma} \Theta_n \Vert_{L^{2}}^{2} \notag\\
& \qquad \leq \Vert \Lambda^{s} \Theta_n \Vert_{L^{2}} \Vert [ \boldsymbol U_{n-1} \cdot \nabla, \Lambda^{s}] \Theta_n  \Vert_{L^{2}} + \Vert \Lambda^{s+\gamma} \Theta_n \Vert_{L^{2}} \Vert \Lambda^{s-\gamma} S \Vert_{L^{2}}  \notag\\
& \qquad \leq C \Vert \Lambda^{s}  \Theta_n \Vert_{L^{2}} \left( \Vert \nabla \boldsymbol U_{n-1} \Vert_{L^{3/\gamma}} \Vert \Lambda^{s} \Theta_n \Vert_{L^{6/(6-\gamma)}} + \Vert \Lambda^{s} \boldsymbol U_{n-1} \Vert_{L^{2}}\Vert  \nabla \Theta_n \Vert_{L^{\infty}}\right) + \Vert \Lambda^{s+\gamma} \Theta_n \Vert_{L^{2}} \Vert \Lambda^{s-\gamma} S \Vert_{L^{2}}  \notag\\
&\qquad \leq C \Vert \Lambda^{s} \Theta_n \Vert_{L^{2}} \Vert \Lambda^{s} \boldsymbol U_{n-1} \Vert_{L^{2}} \Vert \Lambda^{s+\gamma} \Theta_n \Vert_{L^{2}} + \Vert \Lambda^{s+\gamma} \Theta_n \Vert_{L^{2}} \Vert \Lambda^{s-\gamma} S \Vert_{L^{2}}  \notag\\
&\qquad \leq \frac{1}{2} \Vert \Lambda^{s+\gamma} \Theta_n \Vert_{L^{2}}^{2} + C_{s,\gamma} \Vert \Lambda^{s} \boldsymbol U_{n-1} \Vert_{L^{2}}^{2} \Vert \Lambda^{s} \Theta_n \Vert_{L^{2}}^{2} + \Vert \Lambda^{s-\gamma} S \Vert_{L^{2}}^{2} \label{eq:Th:n:bound:1}
\end{align}
where $C_{s,\gamma}> 0$ is a constant. Above we have used the fact that $\nabla \cdot \boldsymbol U_{n-1} = 0$ in order to write the commutator, and also the product and commutator estimates of Proposition~\ref{prop:calculus}. Using the bound \eqref{eq:U:n-1:bound}, and the inductive assumption \eqref{eq:Th:inductive:bound}, it  follows from \eqref{eq:Th:n:bound:1} that
\begin{align}
 \frac{d}{dt} \Vert \Lambda^{s} \Theta_n \Vert_{L^{2}}^{2} + \Vert \Lambda^{s+\gamma} \Theta_n \Vert_{L^{2}}^{2} \leq 4 C_{s,\gamma} C_q \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2 \Vert \Lambda^s \Theta_n \Vert_{L^2}^2 + 2 \Vert \Lambda^{s-\gamma} S \Vert_{L^2}^2
\end{align}
and hence
\begin{align}
  &\Vert \Lambda^{s} \Theta_n(t,\cdot) \Vert_{L^{2}}^{2} + \int_0^t  \Vert \Lambda^{s+\gamma} \Theta_n(\tau,\cdot) \Vert_{L^2}^2 d\tau\notag\\
   & \qquad \leq \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2 e^{4 C_{s,\gamma} C_q \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2 t} + \int_0^t e^{4 C_{s,\gamma} C_q \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2 (t-\tau)} \Vert \Lambda^{s-\gamma} S(\tau,\cdot) \Vert_{L^2}^2 d\tau \notag\\
   & \qquad \leq e^{4 C_{s,\gamma} C_q \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2 t}\left(\Vert \Lambda^s \Theta_0 \Vert_{L^2}^2  + t \Vert S \Vert_{L^\infty(0,\infty;H^{s-\gamma})}^2 \right) \label{eq:Th:n:bound:2}
\end{align}
for all $t\in (0,T)$. Therefore, letting
\begin{align}
  T \leq \min \left\{ \frac{\Vert \Lambda^s \Theta_0 \Vert_{L^2}^2}{2 \Vert S \Vert_{L^\infty(0,\infty;H^{s-\gamma})}^2}, \frac{\ln 4/3}{4 C_{s,\gamma} C_q \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2} \right\} \label{eq:T:choice:1}
\end{align}
we obtain that from \eqref{eq:Th:n:bound:2} that \eqref{eq:Th:inductive:bound} holds for $j=n$, and so by induction it holds for all $j\geq 1$. This shows that the sequence $\Theta_n$ is uniformly bounded in $L^\infty(0,T;H^s) \cap L^2(0,T;H^{s+\gamma})$.

Moreover, we may show that the sequence $\{ \Theta_n \}$ is Cauchy in $L^\infty(0,T;H^{s-1+\gamma})$. To see this, consider the difference of two iterates $\widetilde{\Theta}_n := \Theta_n - \Theta_{n-1}$. It follows from \eqref{eq:Th:n:1} that $\widetilde{\Theta}_n$ is a solution of
\begin{align}
  &\partial_t \widetilde{\Theta}_n + (-\Delta)^\gamma \widetilde{\Theta}_n + \boldsymbol U_{n-1} \cdot \nabla \widetilde{\Theta}_n + \widetilde{\boldsymbol U}_{n-1} \cdot \nabla \Theta_{n-1} = 0 \label{eq:Th:n:diff}\\
  &\widetilde{\Theta}_n(0,\cdot) = 0
\end{align}
for all $n\geq 2$, where we also denoted $\widetilde{\boldsymbol U}_{n} = \boldsymbol M \widetilde{\Theta}_n$. Applying $\Lambda^{s-1+\gamma}$ to \eqref{eq:Th:n:diff}, taking an $L^2$ inner product with $\Lambda^{s-1+\gamma} \widetilde{\Theta}_n$, using $\nabla \cdot \boldsymbol U_{n-1} =0$, and the calculus inequalities of Proposition~\ref{prop:calculus}, we obtain
\begin{align}
  &\frac 12 \frac{d}{dt} \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_n \Vert_{L^2}^2 + \Vert \Lambda^{s-1+ 2\gamma} \widetilde{\Theta}_n \Vert_{L^2}^2 \notag\\
  & \qquad \leq \Vert \Lambda^{s-1 + \gamma} \widetilde{\Theta}_n \Vert_{L^2}  \Vert [ \boldsymbol U_{n-1} \cdot \nabla, \Lambda^{s-1+\gamma}] \widetilde{\Theta}_n \Vert_{L^2} + \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_n \Vert_{L^2}  \Vert \Lambda^{s-1+\gamma} ( \widetilde{\boldsymbol U}_{n-1} \cdot \nabla \Theta_n) \Vert_{L^2}\notag\\
  & \qquad \leq C_{s} \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_n \Vert_{L^2} \left( \Vert \nabla \boldsymbol U_{n-1} \Vert_{L^\infty} \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_n \Vert_{L^2} + \Vert \Lambda^{s-1+\gamma} \boldsymbol U_{n-1} \Vert_{L^6} \Vert \nabla \widetilde{\Theta}_n \Vert_{L^{3}}\right) \notag\\
  & \qquad \qquad + C_{s} \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_n \Vert_{L^2} \left( \Vert \widetilde{\boldsymbol U}_{n-1} \Vert_{L^\infty} \Vert \Lambda^{s +\gamma} \Theta_n \Vert_{L^2} + \Vert \Lambda^{s-1 +\gamma} \widetilde{\boldsymbol U}_{n-1} \Vert_{L^2} \Vert \nabla \Theta_n \Vert_{L^\infty} \right)\notag\\
  & \qquad \leq C_{s,q} \Vert \Lambda^{s-1 +\gamma} \widetilde{\Theta}_n \Vert_{L^2} \left( \Vert \Lambda^{s+\gamma}\Theta_{n-1} \Vert_{L^{2}}\Vert \Lambda^{s-1 +\gamma} \widetilde{\Theta}_n \Vert_{L^2}  + \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_{n-1} \Vert_{L^{2}} \Vert \Lambda^{s +\gamma} \Theta_{n} \Vert_{L^{2}}  \right). \label{eq:Th:Lip:1}
\end{align}
In the last inequality above we have used the assumption $s> 5/2-\gamma$, and the estimate \eqref{eq:U:n-1:bound}. It hence follows from \eqref{eq:Th:Lip:1}, \eqref{eq:Th:inductive:bound}, and the Cauchy-Schwartz inequality that
\begin{align}
\sup_{t \in [0,T]} \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_n (t,\cdot) \Vert_{L^2}
&\leq \sup_{t\in[0,T]} \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_{n-1} (t,\cdot) \Vert_{L^2}
\int_{0}^{T} \Vert \Lambda^{s+\gamma} \Theta_{n}(t,\cdot) \Vert_{L^{2}} e^{\left( \int_{t}^{T} \Vert \Lambda^{s+\gamma} \Theta_{n-1}(\tau,\cdot) \Vert_{L^{2}} d\tau \right) } dt \notag\\
& \leq \sup_{t\in[0,T]} \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_{n-1} (t,\cdot) \Vert_{L^2}
\sqrt{2T} C_{s,q} \Vert \Lambda^{s} \Theta_{0}\Vert_{L^{2}} e^{\sqrt{2T} C_{s,q}\Vert \Lambda^{s} \Theta_{0}\Vert_{L^{2}}} dt.\label{eq:Th:Lip:2}
\end{align}
Recalling \eqref{eq:T:choice:1}, if we let $T$ be such that
\begin{align}
T =\min \left\{ \frac{\Vert \Lambda^s \Theta_0 \Vert_{L^2}^2}{2 \Vert S \Vert_{L^\infty(0,\infty;H^{s-\gamma})}^2}, \frac{\ln 4/3}{4 C_{s,\gamma} C_q \Vert \Lambda^s \Theta_0 \Vert_{L^2}^2}, \frac{C_{s,q}^{2}}{8 \exp(\sqrt{2 \ln 4/3} C_{s,q} / \sqrt{C_{s,\gamma} C_{q}})}  \right\}
\end{align}
we obtain from \eqref{eq:Th:Lip:1} that
\begin{align}
\sup_{t \in [0,T]} \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_n (t,\cdot) \Vert_{L^2}  &\leq \frac{1}{2} \sup_{t\in[0,T]} \Vert \Lambda^{s-1+\gamma} \widetilde{\Theta}_{n-1} (t,\cdot) \Vert_{L^2}.
\end{align}
Thus $\Theta_{n}$ is Cauchy in $L^{\infty}(0,T;H^{s-1+\gamma})$, and hence $\Theta_{n}$ converges strongly to $\Theta$ in $L^{\infty}(0,T;H^{s-1+\gamma})$. Noting that $ s-1+\gamma > 3/2$, this shows that the strong convergence occurs in a H\"older space relative to $x$, which is sufficient to prove that the limiting function $\Theta \in L^{\infty}(0,T;H^{s}) \cap L^{2}(0,T;H^{s+\gamma}) \cap Lip(0,T;H^{\min(s-2\gamma,s-1)}) $ is a solution
of the initial value problem \eqref{eq:1}--\eqref{eq:2}.

To conclude the proof of the theorem, we note that if $\Theta^{(1)}$ and $\Theta^{(2)}$ are two solutions of \eqref{eq:1}--\eqref{eq:2}, then $\Theta = \Theta^{(1)} - \Theta^{(2)}$ solves
\begin{align}
\partial_{t} \Theta + (-\Delta)^{\gamma } \Theta + \boldsymbol U^{(1)} \cdot \nabla \Theta + \boldsymbol U \cdot \nabla \Theta^{(2)} = 0 \label{eq:TH:diff}
\end{align}
with initial condition $\Theta(0,\cdot) = 0$. An $L^{2}$ estimate on \eqref{eq:TH:diff} shows that $\Theta(t,\cdot)= 0 $ for all $t\in[0,t)$, since $\Theta^{(2)} \in L^{\infty}(0,T;H^{s})$ and since if $\Theta^{(1)}$ and $\Theta^{(2)}$ have frequency support on $\mathcal P_{q}$, so does $\Theta$.
\end{proof}

\begin{corollary}[{\bf Local existence for well-prepared data}] \label{cor:well-prepared}
Let $j_{1}, j_{2} \in {\mathbb Z}\setminus\{0\}$, and fix $s,\gamma$ as in the statement of Theorem~\ref{thm:well-prepared:local}. Assume that $\Theta_{0} \in H^{s}({\mathbb T}^{3})$ is such that $\hat{\Theta}_{0}(k_{1},k_{2},k_{3}) \neq 0$ if and only if $(k_{1},k_{2}) = \pm (j_{1},j_{2})$ .  Similarly, assume that  forcing $S\in L^{\infty}(0,T_{s};H^{s-\gamma})$ is such that $\hat{S}(k_{1},k_{2},k_{3},t) \neq 0$ if and only if $(k_{1},k_{2}) = \pm (j_{1},j_{2})$, for all $t\in[0,T_{s})$. Then there exists $T \in (0,T_{s}]$ and a unique solution $\Theta \in L^{\infty}(0,T;H^{s}) \cap L^{2}(0,T;H^{s+\gamma})$ of the initial value problem \eqref{eq:1}--\eqref{eq:2}, and in addition we have that $\hat{\Theta}(k_{1},k_{2},k_{3},t) =0$ whenever $k_{2}/k_{1} \neq j_{2}/j_{1}$.
\end{corollary}

\begin{proof}[Proof of Corollary~\ref{cor:well-prepared}]
Let $q = j_{2}/j_{1} \in \mathbb Q$. The conditions on the initial data and the force imply in particular that their frequency support likes on the plane $\mathcal P_{q}$. The existence of a unique smooth solution $\Theta$, with frequency support lying on $\mathcal P_{q}$ follows directly from Theorem~\ref{thm:well-prepared:local}. But  $(k_{1},k_{2},k_{3}) \in \mathcal P_{q}$ is equivalent to $k_{2} = q k_{1} = j_{2} k_{1} / j_{1}$, so that $\hat{\Theta}(k_{1},k_{2},k_{3}) \neq 0$ only if $k_{2}/k_{1} = j_{2}/j_{1}$.
\end{proof}

\subsection{Global existence and uniqueness for well-prepared data}
\label{sec:well-prepared:global}
In the previous section we have proven that for $\gamma \in (0,1)$, given initial data $\Theta_0$ and source term $S$ which are well-prepared, i.e. they have Fourier support on a plane $\mathcal P_q$ for some $q>0$, then there exists a local in time solution $\Theta$ of the Cauchy problem \eqref{eq:1}--\eqref{eq:3}, which has the property that its Fourier support also lies on $\mathcal P_q$. In this section we show that for $\gamma \in [1/2,1)$ the local in time solution may be continued for all time, while in the case $\gamma \in (0,1/2)$ the same result holds but under the additional assumption that the initial data is small with respect to $\kappa$.

\begin{theorem}[{\bf $ \gamma \geq 1/2$: Global existence for large data}] \label{thm:well-prepared:global}
Let $s, \Theta_0$ and $S$ be as in the statement of Theorem~\ref{thm:well-prepared:local}. If $\gamma \geq 1/2$, the unique smooth solution $\Theta$ of the Cauchy problem  \eqref{eq:FractionalMG:1}--\eqref{eq:FractionalMG:2} is global in time.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:well-prepared:global}]
 Given the conditions on the initial data and of the source term, and using the properties exhibited in Proposition~\ref{prop:support}, the solution constructed in Theorem~\ref{thm:well-prepared:local} satisfies $\mathop {\rm supp} \nolimits(\hat{\Theta}) \subset \mathcal P_{q}$. By Lemma~\ref{lemma:M:order:0} we have that $\boldsymbol U$ is obtained from $\Theta$ by a {\em bounded} Fourier multiplier, and hence by the H\"ormander-Mikhlin theorem we have
 \begin{align}
 \Vert \boldsymbol U \Vert_{W^{s,p}} \leq C \Vert \Theta \Vert_{W^{s,p}}
 \end{align}
 for all $2\leq p < \infty$, and all $s\geq 0$.
Therefore the regime $\gamma > 1/2$ becomes ``sub-critical'' for such solutions, since the map $\Theta\mapsto\boldsymbol U$ is bounded in Sobolev spaces. Therefore, energy estimates which are similar to those used to prove the global regularity of the sub-critical SQG equation (cf.~\cite{ConstantinWu99}), may be used with minor modifications to prove Theorem~\ref{thm:well-prepared:global} in the setting $\gamma > 1/2$.

The case $\gamma = 1/2$ is slightly more delicate. In the case of the critical (SQG) equation, global regularity in the critical case has only been established recently (cf.~\cite{CaffarelliVasseur10, KiselevNazarovVolberg07} ).
Due to the anisotropic nature of the symbol $\boldsymbol M$, it turns out that it is slightly easier to see that the DeGiorgi-inspired proof of \cite{CaffarelliVasseur10} also takes care of the $\gamma=1/2$ case of this theorem. The only fact we must verify is that when $\Theta \in L^\infty$, and $\mathop {\rm supp} \nolimits(\hat \Theta) \in \mathcal P_q$, then\footnote{Note that for generic $\Theta \in L^\infty$ functions $M[\Theta]$ does not belong to $BMO$, but rather to $BMO^{-1}$ (cf.~\cite[Section 4]{FriedlanderVicol11a}).} $\boldsymbol U = \boldsymbol M[\Theta] \in BMO$. Once we prove this, the DeGiorgi iteration scheme of \cite{CaffarelliVasseur10}, shows that $\Theta(t,\cdot)$ is H\"older continuous for $t>0$, and one may use energy arguments to conclude the proof of Theorem~\ref{thm:well-prepared:global}, similarly to \cite[Appendix]{FriedlanderVicol11c}.
Lastly, to verify that $\boldsymbol U \in BMO$,  define a new operator $\boldsymbol M_q$ as the Fourier multiplier with symbol $\hat{\boldsymbol M}(\boldsymbol k) \varphi_q(\boldsymbol k)$, where $\varphi_q(\boldsymbol k)$ is a function which is identically $1$ on $\mathcal P_q$, and vanishes identically at a fixed distance away from $\mathcal P_q$. The advantage is that $\boldsymbol M_q$ is now a periodic pseudo-differential operator of order $0$, and hence maps $L^\infty$ to periodic $BMO$ (cf.~\cite{McLean}). This is attributed to boundedness  of $\hat{\boldsymbol M}_q$, inherited from  the symbols $\boldsymbol M_j$, and follows from the fact that $\mathcal{P}_q$ and its collar neighborhood are not entirely contained in the curved region of the frequency space where $\hat{M}_j$ become unbounded. Lastly one may verify that when $\mathop {\rm supp} \nolimits(\hat{\Theta}) \subset \mathcal P_q$, then $\boldsymbol M_q [\Theta] = \boldsymbol M[\Theta]$, thereby concluding the proof of the theorem.
\end{proof}

\begin{theorem}[{\bf $\gamma < 1/2$: Global existence for small data}] \label{thm:well-prepared:global:a}
Let $\gamma, s, \Theta_0$ and $S$ be as in the statement of Theorem~\ref{thm:well-prepared:local}.
 If $\gamma  <1/2$,  there exists a sufficiently small constant $\varepsilon>0$, such that if
 $\Vert\Theta_0\Vert_{L^2}^\alpha \Vert \Theta_{0} \Vert_{H^{s}}^{1-\alpha} +\Vert\Theta_0\Vert_{L^2}^\alpha \Vert S \Vert_{L^\infty(0,\infty;H^{s-\gamma})}^{1-\alpha}\leq \varepsilon$, where $\alpha = 1 - (5/2-\gamma)/s$, then the unique smooth solution $\Theta$ of the Cauchy problem  \eqref{eq:FractionalMG:1}--\eqref{eq:FractionalMG:2} is global in time.
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{thm:well-prepared:global:a}]
Since $\boldsymbol M$ acts as an operator of order $0$ when applied to functions with frequency support on $\mathcal P_q$, from the point of view of energy estimates the situation we are in is exactly the same as for the super-critical (SQG) equation. The proof the theorem follows from the same arguments used in~\cite{CordobaCordoba04, Ju04, Wu04} to show the global well-posedness of small solutions to the super-critical (SQG) equation.
\end{proof}

redpajama_set_name: RedPajamaArXiv


We are an upholstery shop located in Mid Coast Maine, just minutes from downtown Belfast. For over 35 years we have offered a full range of canvas and upholstery work for marine, residential, commercial, industrial, automotive, and civic applications. We also sell a full line of foam stock along with natural rolled cotton canvas.


text:
Q: jQuery and hyperlink onclick written by PHP I have some ugly PHP code which I will clean up later, but first the code should start working OK. Now it does not execute the post command. In other words, the script find.php will not be executed. Any ideas how to fix this problem? I just would like to use one hyperlink.
<?php

  print "

Play " . $i . "Select

\n";

?>

A: In all honesty, you'd be better off cleaning up your code now. If you check your console, you'll more than likely get an error regarding jQuery not being loaded.
You should create a button with the required information appropriately:
<a href="#" class="mybutton" data-my-id="<?php echo $my; ?>" data-your-id="<?php echo $your; ?>" data-select="<?php echo $i; ?>">Select

And create an event listener in jQuery:
jQuery(document).ready(function() {
    // handle click
    jQuery(document).on('click', 'a.mybutton', function(e) {
        e.preventDefault();
        var data = {
            my: jQuery(this).data('my-id'),
            your: jQuery(this).data('your-id'),
            select: jQuery(this).data('select'),
        }
        //post
        jQuery.post('find.php', data);
        alert(data.select);
        jQuery('html').css({ 'overflow': 'auto', 'height': 'auto' });
     this.parentNode.parentNode.parentNode.parentNode.parentNode.parentNode.style.display = 'none'; return false;

    });
});

From there, you'll have to clean it up to suit exactly what you need. If I were you, I'd start with the correct code/path in mind instead of "looking at it later"

A: It seems you want to create divs dynamically with hyperlinks executing jquery post. You can achieve this entirely in javascript itself  -
function postFunc(my_id, your, i) {
  $.post('find.php', {
    'my': my_id,
    'your': your,
    'select': i
  });
}

function createDynamicDiv(my_id, your, i)
{
  var node = document.createElement('div'); //dynamically creates a div
  node.setAttribute("style", "overflow: auto; height: auto"); //set css
  newlink = document.createElement('a');
  newlink.setAttribute('class', 'dynamicDiv'); //you might want to group all divs in a common class to add common css
  newlink.setAttribute('href', 'javascript:postFunc(my_id,your,i)'); //javascript: prefix on the href makes to interpret postFunc as a javascript function
  node.appendChild(newlink);

}

Call createDynamicDiv from your code to create a dynamic div.

redpajama_set_name: RedPajamaStackExchange


text:
Q: Fetch yahoo mail content within date range using YQL I am using yahoo query language with web service to retrieve yahoo mails using oauth. Now my use case is to download all emails with in given time range.
How can i construct the query to fetch data in time range?
Query i used(it is not working) :
select message from ymail.msgcontent where (mids) in (SELECT  mid  FROM ymail.messages WHERE fid="Sent" and messageInfo.receivedDate >  1393417547

Can someone help me write a right one.

A: Instead of directly selecting mid, select messageInfo.mid will fix it
Below query did the magic :
select message from ymail.msgcontent  where (mids) in (select messageInfo.mid  from ymail.messages where messageInfo.receivedDate > 1392658116)


text:
United Arab Emrites (Parfum)
"I spent three years working in the Middle East where I became fascinated with the odors I encountered. A luxurious exoticism married itself in the air, which was rich, complex, and full of volume. In over thirty years of training, I have never come across a culture which held perfume...
"I spent three years working in the Middle East where I became fascinated with the odors I encountered. A luxurious exoticism married itself in the air, which was rich, complex, and full of volume. In over thirty years of training, I have never come across a culture which held perfume so close to its heart - the ritual of scenting was engrained as part of everyday life. This cast a spell on me; I fell in love with the perfumes of the Middle East and the people who wore them. I created 'The Spirit Of The Union' to capture this devotion, in honor of the nation's deep-rooted appreciation of perfume." – Roja Dove
Notes: Bergamot, Artemisia, Geranium, Rose de Mai, Jasmin de Grasse, Ylang Ylang, Cistus, Clove Buds, Blackcurrant Buds, Pink Pepper, Cumin, Cypriol, Patchouli, Oakmoss, Guaiacwood, Aoud, Benzoin, Frankincense, Labdanum, Musk

redpajama_set_name: RedPajamaCommonCrawl


text:
import { Component } from '@angular/core';
import { Router, ROUTER_DIRECTIVES } from '@angular/router';

import { MdIcon } from '@angular2-material/icon';
import { MD_LIST_DIRECTIVES } from '@angular2-material/list';

import { TD_LAYOUT_DIRECTIVES } from '@covalent/core';

import { SessionService } from '../../services/index'

@Component({
  moduleId: module.id,
  selector: 'qs-main',
  templateUrl: 'main.component.html',
  styleUrls: ['main.component.css'],
  directives: [
    ROUTER_DIRECTIVES,
    MdIcon,
    MD_LIST_DIRECTIVES,
    TD_LAYOUT_DIRECTIVES,
  ],
})
export class MainComponent {

  greetMessage: string

  constructor(private router: Router, private session: SessionService) {
    this.greetMessage = 'Welcome '+session.getUsername();
  }

  routes: Object[] = [{
      title: 'Home',
      route: '/',
      icon: 'home',
    }, {
      title: 'Free Trials and POCs',
      route: '/trials',
      icon: 'cloud',
    }, {
      title: 'App Demo',
      route: '/loader',
      icon: 'event_note',
    }
  ];

  logout(): void {
    this.session.resetSession()
    this.router.navigate(['/login']);
  }
}

redpajama_set_name: RedPajamaGithub


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Durham's Station was located near the site of what was once the headquarters of American Tobacco Company (the world's largest cigarette manufacturer) and is now one of the most ambitious, largest, and farthest-reaching historic preservation and renovation projects in the history of North Carolina. The station was the primary stop for Union officers during surrender negotiations at James Bennett's farm, and Union Commander Kilpatrick established headquarters at the nearby home of Dr. Richard Blacknall. N.C. Civil War marker on Blackwell Street, on the northern end of the Durham Bulls Athletic Park.


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Through independent investment platforms we manage capital for pension funds, sovereign wealth funds, financial institutions and family offices. We build scalable platforms based on thesis-driven strategies. We have a long-term approach to ownership and our commitments. Our investment strategies have a strong sustainability focus.
Our team operates across offices in London, Stockholm and Copenhagen with a representative office in Berlin.
We invest in existing, underperforming assets, which present opportunities for a change through rigorous asset management. Our asset strategies are focused on repositioning, targeted modernisation and enhancement opex and capex programmes.
We also design and develop new properties. We have historically led investment into new property sectors in postindustrial regeneration city areas. We have laid the foundations for new communities to grow and establish their place in the city.
We have delivered best in class multi-family housing, student apartments, micro-living, senior living, serviced apartments, co-living and offices in city cores and in new and emerging neighborhoods.
Our investment approach, coupled with our depth of expertise, enables us to deliver outstanding investment performance across sectors, regions and market cycles.
Our vertically integrated team consists of highly experienced property and investment specialists with a broad set of skills from private equity, investment management, asset management, property development and finance.
This allows us to directly manage the entire investment process, every step of the way from inception to completion: origination, acquisition, financing, repositioning, refurbishment, development, asset management, leasing, marketing, sales and exit. Our service is complete and is tailored to the needs of each investment and asset plan.
The founding partners of Ailon have worked together for over 20 years and have backgrounds from leading international companies such Babcock & Brown Real Estate, Credit Suisse First Boston, BNP Paribas and Richard Ellis.
The team's collective investment volume is over €10 bn across seven jurisdictions, with focus on the UK, the Nordic region and Germany.
Towards the future – Ailon 2030
The world is changing fast and so do people's priorities and expectations. Ailon commits itself to leading the response to these shifts to create the next generation of new, better places for people to live and work.
Delivering better returns is a key principle and we have always endeavoured to 'do the right thing' in our investments and developments. This includes finding optimal ways of delivering and managing our projects, refurbishing and repurposing existing buildings where possible, using offsite modular construction for new developments and integrating energy saving systems and property technology applications into our property management operations. Before it had its present name, we have been promoting the circular economy through our regeneration efforts and we will continue to do so.
We continue to accelerate our ESG initiatives and lead the way towards a carbon neutral real estate sector. We set ourselves the highest standards by placing sustainability at the centre of our operations.
Our asset plans are designed to make the buildings more energy efficient and less carbon intense over its life-cycle. We promote sustainable patterns in both consumption and production/modernisation, reduction of energy requirements and greenhouse gas emissions, to the benefit of our tenants, investors and communities.
Learn more about our commitment to sustainability
Visiting office
Nybrogatan 6
Havnegade 39
239 Kensington High Street
London W8 6SN
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