\begin{theorem}\label{geometricallyFiniteNearOrbit}
    Suppose that a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if there exists an escaping sequence of hyperbolic elements in $G$.
\end{theorem}

\begin{lemma}\label{producingHyperbolicElements}
    Let the action of $G$ on $M$ be a non-elementary convergence group action.
    If there exist an element $f \in G$ and an open set $U \subset M$ such that $f\overline{U} \subsetneqq U$, then $f$ is hyperbolic.
\end{lemma}

Let $X$ be a proper Gromov hyperbolic space, on which a countable group $G$ acts properly by isometries. The action naturally extends to a convergence group action on the Gromov boundary $\partial X$, which decomposes $\partial X$ into the \textit{limit set} $\Lambda G$ and the domain of discontinuity. Let $\Pi \subset \Lambda G$ be the set of parabolic points. The action of $G$ on $X$ is called \textit{geometrically finite} if there exists a sufficiently separated $G$-invariant family of horoballs $\{B_p:p\in \Pi\}$ so that the action on the complement $\mathrm{CH}(\Lambda G)\setminus \bigcup_{p\in \Pi} B_p$ is cocompact, where $CH(\Lambda G)$ denotes the weak convex hull of $\Lambda G$ in $X$. In this case, $\Lambda G\setminus \Pi$ consists precisely of \textit{conical points}. We refer to Section \ref{preliminary} for precise definitions.

Historically, the notion of geometric finiteness originated in the theory of Kleinian groups (see \cite{BeaMas,Marden,thurstonnotes}) and has since been extended to the much broader class of groups considered here in a series of works \cite{Bow3,Bow4,bowditch2012relatively}. Since the formal introduction in \cite{Gromov}, this class of groups has been intensively studied in geometric group theory under the name of \textit{relatively hyperbolic groups}, encompassing many examples of geometric and algebraic origin. Several equivalent characterizations of relative hyperbolicity are then proposed in the literature, generalizing and going well beyond those known for Kleinian groups; see, for example, \cite{Farb, bowditch2012relatively, Osin, DruSapir, Ge1}, as well as \cite{Hru} for an overview and references therein.

The aim of this paper is to supplement these descriptions with two new characterizations in the present general setting.
Let $\{g_n\}_{n\ge 1}$ be a sequence of hyperbolic elements in $G$. We say that $\{g_n\}_{n\ge 1}$ is \textit{escaping} if for any point $o$ and any $R>0$, the $R$-neighborhood of $Go$ intersects the axes of only finitely many $g_n$.

\label{preliminary}
In this section we introduce the preliminaries we shall use on convergence group actions (cf. \cite{tukia1994convergence, bowditch1999convergence}) and hyperbolic spaces (cf. \cite

Let $X$ be a proper Gromov hyperbolic space, on which a countable group $G$ acts properly by isometries. The action naturally extends to a convergence group action on the Gromov boundary $\partial X$, which decomposes $\partial X$ into the \textit{limit set} $\Lambda G$ and the domain of discontinuity. Let $\Pi \subset \Lambda G$ be the set of parabolic points. The action of $G$ on $X$ is called \textit{geometrically finite} if there exists a sufficiently separated $G$-invariant family of horoballs $\{B_p:p\in \Pi\}$ so that the action on the complement $\mathrm{CH}(\Lambda G)\setminus \bigcup_{p\in \Pi} B_p$ is cocompact, where $CH(\Lambda G)$ denotes the weak convex hull of $\Lambda G$ in $X$. In this case, $\Lambda G\setminus \Pi$ consists precisely of \textit{conical points}. We refer to Section \ref{preliminary} for precise definitions.

The aim of this paper is to supplement these descriptions with two new characterizations in the present general setting.
Let $\{g_n\}_{n\ge 1}$ be a sequence of hyperbolic elements in $G$. We say that $\{g_n\}_{n\ge 1}$ is \textit{escaping} if for any point $o$ and any $R>0$, the $R$-neighborhood of $Go$ intersects the axes of only finitely many $g_n$.

The corresponding result for Kleinian groups was proved by Bonahon \cite{bonahon-bouts} and was recently generalized to pinched negatively curved manifolds by Kapovich and Liu \cite{KL19}. In contrast to these works, our approach relies only on elementary arguments in hyperbolic geometry and does not invoke the Margulis lemma.

\begin{theorem}\label{geometricallyFiniteNonconicalCountable}
 Suppose a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if the set of non-conical points is uncountable.
\end{theorem}

Finally, let us conclude the Introduction with an open question.
By a theorem of Yaman \cite{Yaman} (and of Bowditch \cite{Bow98} in the cocompact case), geometrically finite actions on Gromov hyperbolic spaces can be recovered from the dynamics on the boundary. More precisely, if a convergence group action is geometrically finite (i.e., its limit set consists only of conical points and bounded parabolic points) then the group admits a geometrically finite action on some proper Gromov hyperbolic space. Moreover, the induced action on the boundary coincides with the original convergence group action. The following question asks whether the assumption of bounded parabolic points could be weakend as follows.

\begin{question}\label{geometricallyFiniteConvergenceGroup}
 Suppose a discrete group $G$ admits a convergence group action on a compact metrizable space with countably many non-conical limit points. Does $G$ admit a geometrically finite action on some Gromov hyperbolic space?
\end{question}

Together with the standing assumption that the action is non-elementary (which ensures the existence of independent hyperbolic elements), we have a finiteness statement about conjugacy classes.
\begin{lemma}\label{stableTranslationLength}
 For any $r > 0$, there are only finitely many conjugacy classes of hyperbolic elements in $G$ with translation length less than $r$.
\end{lemma}
\begin{proof}
 Let $r > 0$, and let $g \in G$ be a hyperbolic element with $\tau(g) < r$. By our assumption, there exists $h \in G$ and an axis $\gamma$ of $g$ such that $d(h o, \gamma) \le R_0$. Thus, we can choose a point $x \in \gamma$ satisfying $d(h o, x) \le R_0$. Applying the triangle inequality and using Lemma~\ref{translationDistance}, we estimate:
 $$
 d(h o, g h o) \le d(h o, x) + d(x, g x) + d(g x, g h o) \le R_0 + \tau(g) + D_1 + R_0 < 2R_0 + D_1 + r.
 $$
 This yields $d(o, h^{-1} g h \, o) = d(h o, g h o) < 2R_0 + D_1 + r$.
 Since the action of $G$ on $X$ is proper, the set $\{ k \in G : d(o, k o) < 2R_0 + D_1 + r \}$ is finite. Therefore, the set of conjugates $h^{-1}gh$ belongs to this finite set. Consequently, there are only finitely many distinct conjugacy classes of such elements $g$.
\end{proof}

Throughout this section, let $G$ be a discrete group that admits a non-elementary, geometrically infinite, proper action on a proper $\delta$-hyperbolic space $X$ for some $\delta\ge 0$, and fix a basepoint $o\in X$.
By Lemma~\ref{localQuasiGeodesic}, there exist constants $L>16\delta$, $\lambda \ge 1$, and $C\ge 0$ such that every $L$-local $(3, 2D_0)$-quasi-geodesic in $X$ is a $(\lambda, C)$-quasi-geodesic.
By Lemma~\ref{MorseLemma}, there exists a constant $B>0$ such that $d_{\text{Haus}}(p, [p_-, p_+])\le B$ for every $(\lambda, C)$-quasi-geodesic $p$.

\label{preliminary}
In this section we introduce the preliminaries we shall use on convergence group actions (cf. \cite{tukia1994convergence, bowditch1999convergence}) and hyperbolic spaces (cf. \cite

The corresponding result for Kleinian groups was proved by Bonahon \cite{bonahon-bouts} and was recently generalized to pinched negatively curved manifolds by Kapovich and Liu \cite{KL19}. In contrast to these works, our approach relies only on elementary arguments in hyperbolic geometry and does not invoke the Margulis lemma.

Following the strategy of Bishop \cite{Bishop96} for Kleinian groups, we derive the second characterization of geometric infiniteness  from Theorem \ref{geometricallyFiniteNearOrbit}. This refines the result of Beardon and Maskit \cite{BeaMas} in terms of non-conical limit sets.

\begin{theorem}\label{geometricallyFiniteNonconicalCountable}
    Suppose  a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if the set of non-conical points is uncountable.
\end{theorem}

We remark that  the negatively pinched  manifold case was due to  Kapovich and Liu   \cite{KL20} after their work \cite{KL19} on Theorem \ref{geometricallyFiniteNearOrbit}. To put into perspective, for finitely generated geometrically infinite Kleinian groups, the non‑conical limit set has   maximal Hausdorff dimension whose precise value was computed by Bishop–Jones \cite{BJ97bddgeo}, subsequent contributions \cite{KL20}, and recently completed in \cite{mj2025hausdorff} in full general case. The question of whether Theorem~\ref{geometricallyFiniteNonconicalCountable} holds in this level of generality arose naturally in the course of that study.

Finally, let us conclude the Introduction with an open question.
By a theorem of Yaman \cite{Yaman} (and of Bowditch \cite{Bow98} in the cocompact case), geometrically finite actions on Gromov hyperbolic spaces can be recovered from the dynamics on the boundary. More precisely, if a convergence group action is geometrically finite (i.e., its limit set consists only of conical points and bounded parabolic points) then the group admits a geometrically finite action on some proper Gromov hyperbolic space. Moreover, the induced action on the boundary coincides with the original convergence group action. The following question asks whether the assumption of bounded parabolic points could be weakend as follows.

\begin{question}\label{geometricallyFiniteConvergenceGroup}
    Suppose a  discrete group  $G$ admits a convergence group action on a compact metrizable space with  countably many non-conical limit points. Does $G$ admit a geometrically finite action on some Gromov hyperbolic space?
\end{question}

\begin{remark}

\begin{enumerate}
\item
This is related to a question of Kapovich \cite[Problem 7]{Kap05} asking whether every convergence group action can be realized as the boundary action of a proper action on some Gromov hyperbolic space. If so, then a positive answer would follow from Theorem~\ref{geometricallyFiniteNonconicalCountable} together with Yaman’s theorem.
Moreover, in \cite{sun2019dynamicalcharacterizationacylindricallyhyperbolic}, Bin Sun proved that the action constructed in \cite{Bow98} on a hyperbolic space contains WPD (i.e. weak proper discontinuity) elements.

\item The assumption that the action is a convergence group action is necessary. It is easy to give examples of \emph{non-relatively hyperbolic} groups $G$ which admit a minimal action on a compact metriziable space $M$ so that all but {countably many points} $\xi\in M$ are \emph{conical points} in the usual sense.
\begin{itemize}
    \item  there exist a sequence $\{g_n\}$ in $G$ and distinct points $a, b \in M$ such that $g_n x \to b$ locally uniformly for all $x \in M \setminus \{\xi\}$, while $g_n \xi \to a$.
\end{itemize}
Indeed, any non-elementary action on a locally infinite tree $T$ gives raise to the desired action on the compact space $M=\partial T\cup T^0$, the union of the ends of the tree and the vertex set. We equip $M$ with Bowditch topology  (since $T$ is a fine graph in sense of \cite{bowditch2012relatively}). It is easy to check that the ends of the tree are conical points in the above sense.
\end{enumerate}

\end{remark}
\subsection*{Structure of the paper}
Section 2 reviews the necessary preliminaries on convergence group actions and $\delta$-hyperbolic spaces. A key technical lemma (Lemma~\ref{producingHyperbolicElements}) for producing hyperbolic elements under a dynamical condition is presented.
In Section 3, assuming the condition in Theorem~\ref{geometricallyFiniteNearOrbit}, we construct geometric objects analogous to those in a cusp-uniform action. We prove that every point in the limit set $\Lambda G$ is either conical or bounded parabolic, thereby establishing the theorem.
In Section 4, we largely follow the strategy of \cite{Bishop96} to construct uncountably many non-conical limit points under the hypothesis of geometric infiniteness. The proof uses the tool of $L$-local quasi-geodesics to build the required paths, thus proving Theorem~\ref{geometricallyFiniteNonconicalCountable}.

\subsection*{Acknowledgments}
The second-named author is grateful to Mahan MJ  for insightful discussions about  Theorem \ref{geometricallyFiniteNonconicalCountable}.

A proof sketch for Theorem~\ref{geometricallyFiniteNearOrbit} is indicated in the “Structure of the paper” paragraph: in Section~3, “assuming the condition in Theorem~\ref{geometricallyFiniteNearOrbit}, we construct geometric objects analogous to those in a cusp-uniform action” and then “prove that every point in the limit set $\Lambda G$ is either conical or bounded parabolic, thereby establishing the theorem.” The introduction also notes that the approach “relies only on elementary arguments in hyperbolic geometry and does not invoke the Margulis lemma,” and that a “key technical lemma (Lemma~\ref{producingHyperbolicElements}) for producing hyperbolic elements under a dynamical condition is presented” in Section~2.

Question: Let a group $G$ act non-elementarily and properly by isometries on a proper Gromov hyperbolic space $X$. Write $\Lambda G\subset \partial X$ for the limit set, $\Pi\subset \Lambda G$ for the parabolic points, and $CH(\Lambda G)$ for the weak convex hull of $\Lambda G$. The action is called geometrically finite if there exists a sufficiently separated $G$-invariant family of horoballs $\{B_p:p\in \Pi\}$ such that the action on $CH(\Lambda G)\setminus \bigcup_{p\in \Pi} B_p$ is cocompact; geometrically infinite means not geometrically finite. A sequence $\{g_n\}_{n\ge 1}$ of hyperbolic elements of $G$ is called escaping if, for every $o\in X$ and every $R>0$, the $R$-neighborhood of the orbit $Go$ intersects the axes of only finitely many $g_n$. Which of the following statements is equivalent to the action being geometrically infinite?

There exists an escaping sequence of hyperbolic elements in $G$.

Choices:
B. For every sequence $\{g_n\}_{n\ge 1}$ of hyperbolic elements in $G$ whose conjugacy classes are pairwise distinct, the sequence is escaping.
C. There exists a sequence $\{g_n\}_{n\ge 1}$ of hyperbolic elements in $G$ such that, for some $o\in X$ and some $R>0$, the $R$-neighborhood of the orbit $Go$ intersects the axes of only finitely many $g_n$.
D. There exists a sequence $\{g_n\}_{n\ge 1}$ of hyperbolic elements in $G$ such that, for every $o\in X$, there exists $R>0$ for which the $R$-neighborhood of the orbit $Go$ intersects the axes of only finitely many $g_n$.
E. There exists an escaping sequence $\{g_n\}_{n\ge 1}$ of loxodromic isometries of $X$, not necessarily contained in $G$.

existential escaping sequence replaced by universal claim over distinct conjugacy classes

dropped the quantifier requirement 'for every $o$ and every $R>0$'

quantifier order on $R$ weakened from 'for every $R>0$' to 'there exists $R>0$'

sequence no longer required to lie in the acting group $G$

The stem does not explicitly reveal the correct option. It gives the definitions of geometrically finite/infinite and escaping, but does not itself state the equivalence with escaping sequences.

The item is essentially a direct theorem-recall question: it asks which statement is equivalent to geometrical infiniteness, and the correct choice is the theorem statement itself with no substantive reformulation.

Some reasoning is needed to track quantifiers and group-membership conditions across the distractors, especially to reject weakened or strengthened variants. However, the intended answer is still mainly identifiable by recognizing the exact theorem statement rather than generating a deeper conclusion.

The distractors are mathematically plausible and target realistic failure modes: overstrengthening to a universal claim, weakening quantifiers, changing quantifier order, and dropping the requirement that elements lie in G.

A solid theorem-recall MCQ with strong distractors, but it is highly tautological: the correct answer is essentially the exact equivalence being tested, so it offers only moderate generative reasoning.


The evolution of hypersurfaces by their mean curvature has been studied extensively since Huisken \cite{huisken84}, who proved that closed embedded convex hypersurfaces in Euclidean space contract to round points under mean curvature flow.
More recently, the mean curvature flow of submanifolds with higher codimension has been studied. The first author and Baker \cite{andrewsbaker2010} proved an analogous result to that of \cite{huisken84} in any codimension of Euclidean background under a quadratic curvature pinching condition $|h|^2 \leq c |H|^2$ on the initial data, where $h$ is the second fundamental form, $H$ is the mean curvature, and $c$ is a dimensional constant. Since then, Baker and Nguyen \cite{bakernguyen2023} have explored the flow in the case where the background is a sphere of arbitrary codimension, where the sharp quartic pinching condition for the sphere is found in \cite{vogiatzi2024sharpquarticpinchingmean}. Other directions in high codimension have been explored, including ancient solutions \cite{lynchnguyen2021, Risa_2018}, singularity formation \cite{vogiatzi2023singularitymodelshighcodimension}, flow in complex projective space \cite{vogiatzi2023meancurvatureflowhigh}, and flow in pseudo-Euclidean space of submanifolds with maximal spacelike dimension \cite{Lambert_2019}.

In this paper, we consider the mean curvature flow (MCF) of embedded spacelike submanifolds under the assumption that the normal subspace at each point has signature $(1,k)$, so that the codimension may be arbitrary but no two spacelike normal vectors are orthogonal. This implies that the cone of strictly spacelike normal vectors at any chosen point separates into two connected components. In addition, we impose the condition of \emph{spacelike convexity}, which is the condition that the acceleration vector along any geodesic of the submanifold is strictly spacelike, or equivalently that the second fundamental form $h(v,v)$ is strictly spacelike for any non-zero tangent vector $v$. At each point, the values of $h(v,v)$ must lie in the same connected component of the spacelike normal cone, which we call the \emph{inward normal cone}.
 Spacelike-convex submanifolds are a natural analogue of convex hypersurfaces in Euclidean space: For example, the orthogonal projection of such a submanifold onto any maximal spacelike subspace $\R^{n+1,0} \subset \R^{n+1,k}$ is locally convex, and the sectional curvature of the induced metric is positive.
 More interestingly for our purposes, the condition of spacelike convexity leads to a natural pinching condition for the second fundamental form: There exists $\a > 0$ such that $h(v,v) - \a H$ is inward spacelike for all unit $v\in T\S$, and similarly there exists $\b > 0$ such that $h(v,v) - \b H$ lies in the outward normal cone. We call the largest $\a$ and smallest $\b$ for which this holds the inward and outward pinching ratios respectively. Our first result is that such pinching is preserved under the flow.

\begin{proposition}
 Suppose that $\S$ is compact and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF with $F(.,0)$ spacelike and spacelike-convex. Then the inward pinching ratio $\a$ is non-decreasing in $t$, and the outward pinching ratio $\b$ is non-increasing in $t$.
\end{proposition}

In \cite{huisken84}, pinching of the form $h(v,v)\geq \alpha Hg(v,v)$ is preserved under the flow and can be improved via Stampaccia iteration, eventually leading to Huisken's theorem. Such an approach may be possible in our setting, but we instead use an alternative approach using non-collapsing estimates, which has the advantage of giving direct control on the global geometry of the evolving submanifolds. Sheng and Wang \cite{shengwang} proved that embedded mean-convex solutions of mean curvature flow in Euclidean space satisfy a non-collapsing estimate: For any point on the hypersurface there is a touching enclosed ball with curvature comparable to the mean curvature at that point. In \cite{andrews2011noncollapsingmeanconvexmeancurvature}, the first author found a proof of such an estimate using a direct maximum principle argument, which also provides a corresponding result for balls touching from the exterior. This argument enables a very direct proof of the result of \cite{huisken84}. In this paper in Section \ref{noncollapsing section} we prove a version of noncollapsing for space-like convex submanifolds (the first high-codimension setting where such a result is known). This non-collapsing result holds also for the more general class of acausal submanifolds which are spacelike-mean-convex (that is, the mean curvature is strictly spacelike at each point). We say that an acausal spacelike (mean) convex embedding $F: \S^n \to \R^{n+1,k}$ is (interior) noncollapsed with parameter $\delta$ if, for any $x\in \S$, the pseudoball $\left\{|p-F(x)-\delta\frac{H(x)}{|H(x)|^2}|< \frac{\delta}{|H(x)|}\right\}$ has $F(\Sigma)$ in its exterior, touching at $F(x)$. There is a similar definition of exterior noncollapsing. Our second result is that the flow preserves noncollapsing (\cref{noncollapsing section}).

\begin{proposition}
 Suppose that $\S$ is compact and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF with $F(.,0)$ spacelike and spacelike-convex. Then the inward pinching ratio $\a$ is non-decreasing in $t$, and the outward pinching ratio $\b$ is non-increasing in $t$.
\end{proposition}

In \cite{huisken84}, pinching of the form $h(v,v)\geq \alpha Hg(v,v)$ is preserved under the flow and can be improved via Stampaccia iteration, eventually leading to Huisken's theorem. Such an approach may be possible in our setting, but we instead use an alternative approach using non-collapsing estimates, which has the advantage of giving direct control on the global geometry of the evolving submanifolds. Sheng and Wang \cite{shengwang} proved that embedded mean-convex solutions of mean curvature flow in Euclidean space satisfy a non-collapsing estimate: For any point on the hypersurface there is a touching enclosed ball with curvature comparable to the mean curvature at that point. In \cite{andrews2011noncollapsingmeanconvexmeancurvature}, the first author found a proof of such an estimate using a direct maximum principle argument, which also provides a corresponding result for balls touching from the exterior. This argument enables a very direct proof of the result of \cite{huisken84}. In this paper in Section \ref{noncollapsing section} we prove a version of noncollapsing for space-like convex submanifolds (the first high-codimension setting where such a result is known). This non-collapsing result holds also for the more general class of acausal submanifolds which are spacelike-mean-convex (that is, the mean curvature is strictly spacelike at each point). We say that an acausal spacelike (mean) convex embedding $F: \S^n \to \R^{n+1,k}$ is (interior) noncollapsed with parameter $\delta$ if, for any $x\in \S$, the pseudoball $\left\{|p-F(x)-\delta\frac{H(x)}{|H(x)|^2}|< \frac{\delta}{|H(x)|}\right\}$ has $F(\Sigma)$ in its exterior, touching at $F(x)$. There is a similar definition of exterior noncollapsing. Our second result is that the flow preserves noncollapsing (\cref{noncollapsing section}).

The assumption of turning number $1$ is carried out throughout the paper. The Euclidean inner product leads to a notion of long time existence, hence a subsequence convergence result that a suitably rescaled flow admits a subsequence that converges smoothly to a sphere in some maximal spacelike subspace $\R^{n+1,0} \subset \R^{n+1,k}$. The approach to full convergence is motivated by first author's approach on Gauss curvature flow with the idea of considering normalised flow invariant under affine transformations \cite{andrews96} and showing that the original flow is stable near the final shape \cite[Chapter 16]{ben2020}. Since the rescaled flow consists of a Lorentz transform at each subsequent time, we consider the normalised flow that fixes the volume, centre of mass and the maximal spacelike subspace that has the minimum projected volume. This leads to a stability argument which proves our final result: an analogue of Huisken's theorem.

\begin{proposition}[spacelike-convexity is preserved]\label{spacelike-convexity is preserved}
 Suppose that $\S$ is compact, spacelike-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then spacelike-convexity is preserved along the flow.
\end{proposition}

\begin{proposition}[Pinching is preserved]\label{pinching preserved prop}
 Suppose that $\S$ is compact, spacelike-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then there exists some $\a > 0$ such that $h(v,v) - \a H$ is spacelike for all unit $v\in T\S$ at all times.
\end{proposition}

First, we show that spacelike-mean-convexity is preserved, assuming that the induced metric stays spacelike, this condition is implied by noncollapsing so we omit the assumption for results in this section.
\begin{proposition}
 Suppose that $\S$ is compact, spacelike-mean-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then spacelike-mean-convexity is preserved along the flow.
\end{proposition}

\begin{theorem}\label{acausal spacelike-mean-convex submanifold preserves noncollapsing}
 Suppose $F: \S^n \times [0, T) \to \R^{n+1,k}$ is a spacelike-convex, compact and embedded solution to MCF, if $\S\times \{0\}$ is $\d_-$-exterior noncollapsed and $\d_+$-interior noncollapsed, then
 \begin{equation*}
 \overline{Q}(x, \d_-, t) \leq 0, \qquad \underline{Q}(x,\d_+, t) \geq 0
 \end{equation*}
 for all $t\in [0, T)$.
\end{theorem}

\begin{corollary}\label{spacelike-mean-convex data stays spacelike coro}
 Suppose that $\S$ is compact, spacelike-mean-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then $\S_t$ stays spacelike under the flow.
\end{corollary}

\begin{proposition}[Short time existence]\label{short time existence MCF}
    Suppose that $\S$ is compact, spacelike-convex and let $F_0: \S^n \to \R^{n+1,k}$ be a smooth immersion, then there exists $\e > 0$ and a smooth solution $F(\cdot,t): \S^n \to \R^{n+1,k}$ to MCF defined for $t\in [0, \e)$ satisfying $F(\cdot,0) = F_0$.
\end{proposition}

\section{Noncollapsing}\label{noncollapsing section}

The result holds for a more general class of submanifolds, which is an analogue of mean-convex hypersurfaces.

\begin{definition}
    A spacelike submanifold $\S^n \subset \R^{n+1,k}$ is \emph{spacelike-mean-convex} if its mean curvature is everywhere spacelike and non-vanishing.
\end{definition}

First, we show that spacelike-mean-convexity is preserved, assuming that the induced metric stays spacelike, this condition is implied by noncollapsing so we omit the assumption for results in this section.
\begin{proposition}
    Suppose that $\S$ is compact, spacelike-mean-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then spacelike-mean-convexity is preserved along the flow.
\end{proposition}

The assumption of turning number $1$ is carried out throughout the paper. The Euclidean inner product leads to a notion of long time existence, hence a subsequence convergence result that a suitably rescaled flow admits a subsequence that converges smoothly to a sphere in some maximal spacelike subspace $\R^{n+1,0} \subset \R^{n+1,k}$. The approach to full convergence is motivated by first author's approach on Gauss curvature flow with the idea of considering normalised flow invariant under affine transformations \cite{andrews96} and showing that the original flow is stable near the final shape \cite[Chapter 16]{ben2020}. Since the rescaled flow consists of a Lorentz transform at each subsequent time, we consider the normalised flow that fixes the volume, centre of mass and the maximal spacelike subspace that has the minimum projected volume. This leads to a stability argument which proves our final result: an analogue of Huisken's theorem.

\begin{theorem}
    Let $F: \S^n \times [0, T) \to \R^{n+1,k}$ be a maximal solution to MCF, if $F(\cdot, 0)$ is a spacelike-convex embedding, then $F(\cdot, t)$ is a spacelike-convex embedding for all $0 < t < T $, $F_t$ converges uniformly to a point as $t \to T$. Moreover, a suitably rescaled flow $\Tilde{F}(\cdot, \tau)$ exists for all $\tau \in [0, \infty)$ and converges in smooth topology to a smooth embedding whose image coincides with $S^n \subset \R^{n+1,0}$ of $\R^{n+1,k}$.
\end{theorem}

For $n = 1$, this is an analogue of Gage--Hamilton theorem.

The post-theorem introduction does not sketch a proof of the stated noncollapsing preservation result. It instead outlines an approach toward a later “final result” (an analogue of Huisken's theorem): using the Euclidean inner product to obtain “long time existence” and a “subsequence convergence result” for a suitably rescaled flow converging smoothly to a sphere in a maximal spacelike subspace; then, to upgrade subsequence convergence to full convergence, it follows an approach “motivated by” work on Gauss curvature flow by considering a “normalised flow invariant under affine transformations” and proving the “original flow is stable near the final shape.” Because the rescaled flow involves “a Lorentz transform at each subsequent time,” it instead considers a normalised flow that “fixes the volume, centre of mass and the maximal spacelike subspace that has the minimum projected volume,” leading to “a stability argument” establishing the Huisken-type convergence.

Suppose that $\S$ is compact, spacelike (mean) convex and acausal, and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, and for $n = 1$ we require the curve to have turning number (degree of the Gauss map) $1$ , then noncollapsing is preserved.,

Question:
Let $F:\Sigma^n\times I\to\mathbb{R}^{n+1,k}$ be a solution of mean curvature flow $(\partial_tF=H)$ for a compact acausal spacelike submanifold, where acausal means that no two points of the submanifold are timelike-separated in the ambient pseudo-Euclidean space $\mathbb{R}^{n+1,k}$. Assume the submanifold is spacelike (mean) convex: either it is spacelike-convex, meaning $h(v,v)$ is strictly spacelike for every nonzero tangent vector $v$, or it is spacelike-mean-convex, meaning the mean curvature vector $H$ is everywhere spacelike and nonzero. If $n=1$, also assume the curve has turning number (degree of the Gauss map) $1$. For $\delta>0$, say a time-slice is $\delta$-interior noncollapsed if for every $x\in\Sigma$ the pseudoball
$$\left\{p:\left|p-F(x)-\delta\frac{H(x)}{|H(x)|^2}\right|<\frac{\delta}{|H(x)|}\right\}$$
touches the submanifold at $F(x)$ and the image lies outside it; define $\delta$-exterior noncollapsed analogously using the corresponding exterior touching pseudoball. Which statement holds for every such flow?

Noncollapsing is preserved along the flow: if the initial slice is noncollapsed (in particular, if it is $\delta_+$-interior noncollapsed and/or $\delta_-$-exterior noncollapsed), then every later time-slice remains noncollapsed with the same corresponding noncollapsing parameter(s) for all times in $I$.

Choices:
B. Noncollapsing improves along the flow: if the initial slice is noncollapsed, then for every later time-slice there exists a parameter strictly larger than the initial interior noncollapsing parameter and strictly larger than the initial exterior noncollapsing parameter for which the corresponding interior and exterior noncollapsing inequalities hold throughout $I$.
C. One-sided noncollapsing is preserved along the flow: if the initial slice is $\delta_+$-interior noncollapsed, then every later time-slice remains $\delta_+$-interior noncollapsed for all times in $I$; likewise, if the initial slice is $\delta_-$-exterior noncollapsed, then every later time-slice remains $\delta_-$-exterior noncollapsed.
D. Noncollapsing holds automatically along the flow from the geometric assumptions alone: every compact acausal spacelike (mean) convex solution, with the turning-number-$1$ hypothesis when $n=1$, is both $\delta_+$-interior noncollapsed and $\delta_-$-exterior noncollapsed for some positive constants $\delta_+,\delta_-$ depending only on $n$ and $k$, for all times in $I$, without any initial noncollapsing assumption.
E. Noncollapsing is preserved along the flow provided one strengthens the initial hypothesis to two-sided noncollapsing: if the initial slice is simultaneously $\delta$-interior noncollapsed and $\delta$-exterior noncollapsed for the same parameter $\delta>0$, then every later time-slice remains both $\delta$-interior and $\delta$-exterior noncollapsed for all times in $I$; however, preservation need not hold for only one of the two sides separately.

The stem does not explicitly reveal the correct conclusion. It states the hypotheses and definitions, but the exact preservation statement in choice A is not directly given in the prompt.

The item is essentially a direct theorem-recall question: the stem lists the assumptions of a preservation theorem and asks for the conclusion. Choice A is basically the theorem statement itself.

There is some reasoning pressure because the options differ by subtle quantifier and strength changes (one-sided vs two-sided, same parameter vs weakened parameter). However, the task is mostly recognition of the exact theorem rather than genuine derivation or synthesis.

The distractors are plausible and mathematically relevant: they reflect common misreadings such as requiring both sides initially, allowing parameter degradation, or preserving only one side. They are distinct and well targeted.

A solid recognition-style theorem question with strong distractors, but it is highly tautological and only moderately tests reasoning rather than generative understanding.


\begin{eqnarray}\label{eq:dz-w}
		 \dz w &=&-\text{div}_h\tilde v+\frac{\gamma-1}{\gamma p}\left(\kappa\partial_z^2\theta+\widetilde{Q(\nabla v)}\right)-\frac{1}{\gamma p}\tilde v\cdot\nabla_hp\nonumber\\
&=&\dfrac{(\gamma-1)\kappa}{\gamma R}\partial_{zz} \sigma -\text{div}_h\tilde v-\frac{1}{\gamma p}\left(\tilde v\cdot\nabla_hp-(\gamma-1)\widetilde{Q(\nabla v)}\right)\nonumber\\
&=&\nu\partial_z^2\sigma-\phi(v,p),\label{EQw00}
\end{eqnarray}

\begin{eqnarray}
	&\mathbb S_h = \mu (\nablah v + \nablah^\top v) + \lambda \dvh v \mathbb I_2, \quad
	p = R \rho \theta, \nonumber\\
  &Q(\nabla v)=\mathbb S_h:\nabla_hv+\mu|\partial_zv|^2,\label{Q}
\end{eqnarray}

\begin{eqnarray}
  &\partial_tv+(v\cdot\nabla_h)v+w\partial_zv+\sigma\nabla_hp=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\text{div}_hv,\label{EQv}\\
  &w=\nu\partial_zv-\int_0^z\phi(v,p)dz',\label{EQw}\\
  &\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma(\phi(v,p)-\text{div}_hv)=\nu\sigma\partial_z^2\sigma+\epsilon\Delta_h\sigma,
  \label{EQs}\\
  &\partial_tp+\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v=\epsilon\Delta_hp+(\gamma-1)\overline{Q(\nabla v)}, \label{EQp}
\end{eqnarray}

\begin{equation}\label{eq:CPE}
	\begin{cases}
		\dt \rho + \dvh (\rho v)  + \dz (\rho w) = 0, \\
		\dt (\rho v) + \dvh(\rho v\otimes v) + \dz(\rho w v) + \nablah p = \dvh \mathbb S_h + \mu \partial_{zz} v, \\
		\dz p = 0, \\
		\dfrac{1}{\gamma-1}\bigl(\dt p + \dvh(p v) + \dz (pw) \bigr) + p(\dvh v + \dz w) = \dz (\kappa \dz \theta )+Q(\nabla v),
	\end{cases}
\end{equation}

\begin{theorem}\label{thmmain}
Given $v_{0}\in H^3(\mathcal O)$, $\sigma_{0} \in H^2(\mathcal O)$, and $p_{0}\in H^3(\mathbb T^2),$ such that
$$
\sigma_{0}\geq\underline\sigma, \quad p_{0}\geq\underline p,\quad \partial_zv_{0}|_{z=0,1}=0,\quad \partial_z\sigma_{0}|_{z=0,1}=0,
$$
for two positive numbers $\underline\sigma$ and $\underline p$.
Then, there is a positive time $\mathcal T_0$,
depending only on $\gamma$, $\nu$, $\mu$, $\lambda$, $\underline\sigma$, $\underline p$, and
  $\|v_{0}\|_{H^3}^2+\|\sigma_{0}\|_{H^2}^2+\|p_0\|_{H^3}^2$, such that system (\ref{EQv0})--(\ref{EQp0}),
  subject to (\ref{BC1})--(\ref{IC}), has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$,
  satisfying
  \begin{eqnarray*}
  &&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\geq0.5\underline\sigma,\quad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\geq0.5\underline p,\\
    &&v\in C([0,\mathcal T_0]; H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0; H^3(\mathcal O))\cap L^2(0,\mathcal T_0; H^4(\mathcal O)),\\
    &&\sigma\in C([0,\mathcal T_0]; H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0; H^2(\mathcal O)),\quad \partial_z\sigma\in L^2(0,\mathcal T_0; H^2(\mathcal O)),\\
    &&\partial_tv\in L^2(0,\mathcal T_0; H^2(\mathcal O)),\quad \partial_t\sigma\in L^2(0,\mathcal T_0; H^1(\mathcal O)),\\
    &&p\in C([0,\mathcal T_0]; H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0; H^3(\mathbb T^2)),\quad\partial_tp\in  L^2(0,\mathcal T_0; H^2(\mathbb T^2)).
  \end{eqnarray*}
\end{theorem}

\begin{equation}\label{bc:CPE}
	\dz \theta\big|_{z=0,1} = 0, \quad \dz v\big|_{z=0,1} = 0, \quad w\big|_{z=0,1} = 0.
\end{equation}

The compressible primitive equations, formally obtained as the hydrostatic approximation of the compressible Navier-Stokes equations by replacing the vertical momentum balance equation with the hydrostatic balance
equation, are the fundamental equations for modern meteorological study (see, e.g., \cite[Chapter 4]{Richardson1965}). In particular, it is the starting point of many large scale
models in the theoretical investigations and practical weather predictions (see, e.g., \cite{Lions1992,JLLions1992,Washington2005}). In fact, such an approximation is reliable and useful in the following sense: (i)
the vertical scale of the atmosphere is significantly smaller than the planetary horizontal scale; (ii) the balance of gravity and pressure dominates the dynamic in the vertical direction; (iii) the vertical velocity is
usually hard to observe in reality. Rigorous justifications of the hydrostatic approximation of hydrodynamic equations can be found in \cite{Li2017,Azerad2001} (see also \cite{FugigaHHK,FuGiHiHuKaW1,FuruGigaKas,LiTitiYuan}) for incompressible flows and \cite{Liu2019} for
isentropic compressible flows.

In this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:
\begin{equation}\label{eq:CPE}
	\begin{cases}
		\partial_t \rho + \mathrm{div}_{h}\, (\rho v)  + \partial_z (\rho w) = 0, \\
		\partial_t (\rho v) + \mathrm{div}_{h}\,(\rho v\otimes v) + \partial_z(\rho w v) + \nabla_{h} p = \mathrm{div}_{h}\, \mathbb S_h + \mu \partial_{zz} v, \\
		\partial_z p = 0, \\
		\dfrac{1}{\gamma-1}\bigl(\partial_t p + \mathrm{div}_{h}\,(p v) + \partial_z (pw) \bigr) + p(\mathrm{div}_{h}\, v + \partial_z w) = \partial_z (\kappa \partial_z \theta )+Q(\nabla v),
	\end{cases}
\end{equation}
with
\begin{eqnarray}
	&\mathbb S_h = \mu (\nabla_{h} v + \nabla_{h}^\top v) + \lambda \mathrm{div}_{h}\, v \mathbb I_2, \quad
	p = R \rho \theta, \nonumber\\
  &Q(\nabla v)=\mathbb S_h:\nabla_hv+\mu|\partial_zv|^2,\label{Q}
\end{eqnarray}
with constant coefficients $\gamma$, $\mu$, $\lambda$, $\kappa,$ and $R$ satisfying $\gamma > 1$, $\mu+\lambda>0$, $\mu>0$, $\kappa>0$, and $R>0$.
Here, $ \rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \mathrm{div}_{h}\,, \nabla_{h} $, and $\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.
We investigate system \eqref{eq:CPE} in the horizontally periodic channel
$$ \mathcal O:= \mathbb T^2 \times (0,1) = \lbrace (x,y,z) | (x,y) \in \mathbb T^2, z \in (0,1) \rbrace,
$$
and consider the following
boundary conditions:
\begin{equation}\label{bc:CPE}
	\partial_z \theta\big|_{z=0,1} = 0, \quad \partial_z v\big|_{z=0,1} = 0, \quad w\big|_{z=0,1} = 0.
\end{equation}

The isentropic compressible primitive equations have been investigated by the last two authors in \cite{LT2018a} for local strong solutions and \cite{LT2018b} for global weak solutions. The existence of global weak solutions is also studied by \cite{Wang2020}, independently. See also, \cite{Jiu2018,Ersoy2011a,Ersoy2012,Gatapov2005}.

On the other hand, the incompressible primitive equations have been the subject of intensive mathematical research since the introduction
by Lions, Temam, and Wang in \cite{Lions1992,JLLions1992}. For instance, Guill\'en-Gonz\'alez, Masmoudi, and Rodr\'iguez-Bellido in
\cite{GuillenGonzalez2001} study the local existence of strong solutions and global existence of strong solutions with small initial data. In
\cite{HuTemamZiane2003}, the authors address the global existence of strong solutions in a domain with small depth. The first breakthrough
concerning the global well-posedness of incompressible primitive equations is obtained by Cao and Titi in \cite{Cao2007}. See also,
\cite{Cao2014a,Cao2014b,Cao2016,Cao2016a,CaoLiTiti2020,CaoLiTitiWang2024,Boling,MHATK,MHTKa,GMK,IKMZ,Li2017a,LiYuan2022} and the references therein for
related literatures; in particular, global well-posedness of strong solutions was proved in \cite{Cao2016,Cao2016a,CaoLiTiti2020,CaoLiTitiWang2024} to the primitive equations with only horizontal viscosity.

We have introduced the PE diagram in \cite{LT2018LowMach1}, which concerns the low Mach number limit and the small aspect ratio (between the vertical and horizontal scales) limit. In \cite{LT2018LowMach1,LT2018LowMach2}, we also establish the low Mach number limit for the isentropic compressible primitive equations. However, the counterpart study of the compressible Navier-Stokes-Fourier equations (see, e.g., \cite{Alazard2006}) indicates that the PE diagram might be completely different for non-isentropic flows, due to the additional heat conductivity. We leave such a subject to future study.

For interested readers, we refer to \cite{Temam1977,Lions1996,Lions1998,Feireisl2004,Feireisl2009a} for the study of hydrodynamic equations.

One can easily see that, compared to the Navier-Stokes equations, the evolutionary equation for vertical velocity is missing in system
\eqref{eq:CPE}. In fact, this is one of the main challenges in the study of the compressible primitive equations. In order to have a better
understanding of the system, we derive a representation of vertical velocity in terms of horizontal velocity, density, and pressure.
Consequently, we will reformulate system \eqref{eq:CPE} to an equivalent one.

Use \subeqref{eq:CPE}{1} to rewrite \subeqref{eq:CPE}{2} as
$$
\rho(\partial_tv+(v\cdot\nabla_h)v+w\partial_zv)+\nabla_hp=\mu\Delta v+(\mu+\lambda)\nabla_h\text{div}_hv
$$
and define
\begin{equation}\label{def:rho-1}
	\sigma := \frac1\rho.
\end{equation}
Then, it is clear that
\begin{equation}
\partial_tv+(v\cdot\nabla_h)v+w\partial_zv+\sigma \nabla_hp=\mu\sigma\Delta v+(\mu+\lambda)\sigma \nabla_h\text{div}_hv. \label{EQv00}
\end{equation}

Note that \subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \subeqref{eq:CPE}{3} to rewrite \subeqref{eq:CPE}{4} as
\begin{equation*}
 \partial_tp+v\cdot\nabla_hp+\gamma p(\text{div}_hv+\partial_zw)=(\gamma-1)\kappa\partial_z^2\theta+(\gamma-1)Q(\nabla v).
\end{equation*}
Separating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\partial_z\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that
\begin{eqnarray}
 \partial_tp+\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)},\label{EQp00}\\
\tilde v\cdot\nabla_hp+\gamma p(\text{div}_h\tilde v+\partial_zw)=(\gamma-1)\kappa\partial_z^2\theta+(\gamma-1)\widetilde{Q(\nabla v)}. \label{EQw000}
\end{eqnarray}
Here, for a function $f$, we use
\begin{equation}\label{TBf}
\bar f:=\int_0^1f dz\quad\mbox{and}\quad  \tilde f:=f-\bar f,
\end{equation}
to represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.

Since $\theta = R^{-1} \rho^{-1} p=R^{-1}\sigma p,$
it follows from \eqref{EQw000} and \subeqref{eq:CPE}{3} that
\begin{eqnarray}\label{eq:dz-w}
		 \partial_z w &=&-\text{div}_h\tilde v+\frac{\gamma-1}{\gamma p}\left(\kappa\partial_z^2\theta+\widetilde{Q(\nabla v)}\right)-\frac{1}{\gamma p}\tilde v\cdot\nabla_hp\nonumber\\
&=&\dfrac{(\gamma-1)\kappa}{\gamma R}\partial_{zz} \sigma -\text{div}_h\tilde v-\frac{1}{\gamma p}\left(\tilde v\cdot\nabla_hp-(\gamma-1)\widetilde{Q(\nabla v)}\right)\nonumber\\
&=&\nu\partial_z^2\sigma-\phi(v,p),\label{EQw00}
\end{eqnarray}
where $\nu=\dfrac{(\gamma-1)\kappa}{\gamma R}$ and
\begin{eqnarray}
  \phi(v,p):=\text{div}_h\tilde v+\frac{1}{\gamma p}\left(\tilde v\cdot\nabla_hp-(\gamma-1)\widetilde{Q(\nabla v)}\right).\label{phi}
\end{eqnarray}
Recalling that $p$ is independent of $z$, it holds that $\partial_z\sigma=R\partial_z(p^{-1}\theta)= R p^{-1}\partial_z\theta$. Then, the boundary condition \eqref{bc:CPE} implies
\begin{equation}\label{bc:rf-CPE}
\partial_z \sigma\big|_{z=0,1} = 0.
\end{equation}
Thanks to this and recalling that $w|_{z=0}$, it follows from \eqref{EQw00} that
\begin{equation}
  w=\nu\partial_z\sigma-\int_0^z\phi(v,p)dz'.\label{EQw00'}
\end{equation}

It follows from \subeqref{eq:CPE}{1} that
\begin{equation*}\label{eq:rho-1}
	\partial_t \rho^{-1} + v \cdot \nabla_{h} \rho^{-1} - \rho^{-1} \mathrm{div}_{h}\, v = \rho^{-2} \partial_z (\rho w) = \rho^{-1} \partial_z w - w \partial_z \rho^{-1},
\end{equation*}
from which, using \eqref{eq:dz-w} and recalling $\sigma=\frac1\rho$, one gets
\begin{equation}\label{EQs00}
	\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma(\phi(v,p)-\text{div}_hv)=\nu\sigma\partial_z^2\sigma.
\end{equation}

Now, collecting \eqref{EQv00}, \eqref{EQp00}, \eqref{EQw00'}, \eqref{EQs00}, we have the following reformulated system which is equivalent to \eqref{eq:CPE}
\begin{eqnarray}
  &\partial_tv+(v\cdot\nabla_h)v+w\partial_zv+\sigma\nabla_hp=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\text{div}_hv,\label{EQv0}\\
  &w=\nu\partial_z\sigma-\int_0^z\phi(v,p)dz',\label{EQw0}\\
  &\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma(\phi(v,p)-\text{div}_hv)=\nu\sigma\partial_z^2\sigma,
  \label{EQs0}\\
  &\partial_tp+\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)}, \label{EQp0}
\end{eqnarray}
where $\phi(v,p)$ and $Q(\nabla v)$ are expressed as in (\ref{phi}) and (\ref{Q}), respectively. The boundary conditions read as
\begin{eqnarray}
  &v, \sigma, \mbox{ and } p \mbox{ are periodic in }x, y, \label{BC1}\\
  &\partial_zv|_{z=0,1}=0, \quad\partial_z\sigma|_{z=0,1}=0. \label{BC2}
\end{eqnarray}
The initial condition is
\begin{equation}
  \label{IC}
  (v, \sigma, p)|_{t=0}=(v_{0}, \sigma_{0}, p_{0}).
\end{equation}

Before stating the main result, we recall some standard notations. For positive integer $k$ and $q\in[1,\infty]$, $L^q(\mathcal O)$ and
$L^q(\mathbb T^2)$ are the Lebesgue spaces,
$W^{k,q}(\mathcal O)$ and $W^{k,q}(\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,
we will use $\|\cdot\|_q$ to denote $L^q(\mathcal O)$ or $L^q(\mathbb T^2)$ norms of the corresponding function,
if the meaning is clear from the context.

We are now in the position to state the main result of this paper.

In this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:
\begin{equation}\label{eq:CPE}
 \begin{cases}
 \partial_t \rho + \mathrm{div}_{h}\, (\rho v) + \partial_z (\rho w) = 0, \\
 \partial_t (\rho v) + \mathrm{div}_{h}\,(\rho v\otimes v) + \partial_z(\rho w v) + \nabla_{h} p = \mathrm{div}_{h}\, \mathbb S_h + \mu \partial_{zz} v, \\
 \partial_z p = 0, \\
 \dfrac{1}{\gamma-1}\bigl(\partial_t p + \mathrm{div}_{h}\,(p v) + \partial_z (pw) \bigr) + p(\mathrm{div}_{h}\, v + \partial_z w) = \partial_z (\kappa \partial_z \theta )+Q(\nabla v),
 \end{cases}
\end{equation}
with
\begin{eqnarray}
 &\mathbb S_h = \mu (\nabla_{h} v + \nabla_{h}^\top v) + \lambda \mathrm{div}_{h}\, v \mathbb I_2, \quad
 p = R \rho \theta, \nonumber\\
 &Q(\nabla v)=\mathbb S_h:\nabla_hv+\mu|\partial_zv|^2,\label{Q}
\end{eqnarray}
with constant coefficients $\gamma$, $\mu$, $\lambda$, $\kappa,$ and $R$ satisfying $\gamma > 1$, $\mu+\lambda>0$, $\mu>0$, $\kappa>0$, and $R>0$.
Here, $ \rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \mathrm{div}_{h}\,, \nabla_{h} $, and $\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.
We investigate system \eqref{eq:CPE} in the horizontally periodic channel
$$ \mathcal O:= \mathbb T^2 \times (0,1) = \lbrace (x,y,z) | (x,y) \in \mathbb T^2, z \in (0,1) \rbrace,
$$
and consider the following
boundary conditions:
\begin{equation}\label{bc:CPE}
 \partial_z \theta\big|_{z=0,1} = 0, \quad \partial_z v\big|_{z=0,1} = 0, \quad w\big|_{z=0,1} = 0.
\end{equation}

Note that \subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \subeqref{eq:CPE}{3} to rewrite \subeqref{eq:CPE}{4} as
\begin{equation*}
 \partial_tp+v\cdot\nabla_hp+\gamma p(\text{div}_hv+\partial_zw)=(\gamma-1)\kappa\partial_z^2\theta+(\gamma-1)Q(\nabla v).
\end{equation*}
Separating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\partial_z\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that
\begin{eqnarray}
 \partial_tp+\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)},\label{EQp00}\\
\tilde v\cdot\nabla_hp+\gamma p(\text{div}_h\tilde v+\partial_zw)=(\gamma-1)\kappa\partial_z^2\theta+(\gamma-1)\widetilde{Q(\nabla v)}. \label{EQw000}
\end{eqnarray}
Here, for a function $f$, we use
\begin{equation}\label{TBf}
\bar f:=\int_0^1f dz\quad\mbox{and}\quad \tilde f:=f-\bar f,
\end{equation}
to represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.

Since $\theta = R^{-1} \rho^{-1} p=R^{-1}\sigma p,$
it follows from \eqref{EQw000} and \subeqref{eq:CPE}{3} that
\begin{eqnarray}\label{eq:dz-w}
 \partial_z w &=&-\text{div}_h\tilde v+\frac{\gamma-1}{\gamma p}\left(\kappa\partial_z^2\theta+\widetilde{Q(\nabla v)}\right)-\frac{1}{\gamma p}\tilde v\cdot\nabla_hp\nonumber\\
&=&\dfrac{(\gamma-1)\kappa}{\gamma R}\partial_{zz} \sigma -\text{div}_h\tilde v-\frac{1}{\gamma p}\left(\tilde v\cdot\nabla_hp-(\gamma-1)\widetilde{Q(\nabla v)}\right)\nonumber\\
&=&\nu\partial_z^2\sigma-\phi(v,p),\label{EQw00}
\end{eqnarray}
where $\nu=\dfrac{(\gamma-1)\kappa}{\gamma R}$ and
\begin{eqnarray}
 \phi(v,p):=\text{div}_h\tilde v+\frac{1}{\gamma p}\left(\tilde v\cdot\nabla_hp-(\gamma-1)\widetilde{Q(\nabla v)}\right).\label{phi}
\end{eqnarray}
Recalling that $p$ is independent of $z$, it holds that $\partial_z\sigma=R\partial_z(p^{-1}\theta)= R p^{-1}\partial_z\theta$. Then, the boundary condition \eqref{bc:CPE} implies
\begin{equation}\label{bc:rf-CPE}
\partial_z \sigma\big|_{z=0,1} = 0.
\end{equation}
Thanks to this and recalling that $w|_{z=0}$, it follows from \eqref{EQw00} that
\begin{equation}
 w=\nu\partial_z\sigma-\int_0^z\phi(v,p)dz'.\label{EQw00'}
\end{equation}

Now, collecting \eqref{EQv00}, \eqref{EQp00}, \eqref{EQw00'}, \eqref{EQs00}, we have the following reformulated system which is equivalent to \eqref{eq:CPE}
\begin{eqnarray}
 &\partial_tv+(v\cdot\nabla_h)v+w\partial_zv+\sigma\nabla_hp=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\text{div}_hv,\label{EQv0}\\
 &w=\nu\partial_z\sigma-\int_0^z\phi(v,p)dz',\label{EQw0}\\
 &\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma(\phi(v,p)-\text{div}_hv)=\nu\sigma\partial_z^2\sigma,
 \label{EQs0}\\
 &\partial_tp+\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)}, \label{EQp0}
\end{eqnarray}
where $\phi(v,p)$ and $Q(\nabla v)$ are expressed as in (\ref{phi}) and (\ref{Q}), respectively. The boundary conditions read as
\begin{eqnarray}
 &v, \sigma, \mbox{ and } p \mbox{ are periodic in }x, y, \label{BC1}\\
 &\partial_zv|_{z=0,1}=0, \quad\partial_z\sigma|_{z=0,1}=0. \label{BC2}
\end{eqnarray}
The initial condition is
\begin{equation}
 \label{IC}
 (v, \sigma, p)|_{t=0}=(v_{0}, \sigma_{0}, p_{0}).
\end{equation}

Before stating the main result, we recall some standard notations. For positive integer $k$ and $q\in[1,\infty]$, $L^q(\mathcal O)$ and
$L^q(\mathbb T^2)$ are the Lebesgue spaces,
$W^{k,q}(\mathcal O)$ and $W^{k,q}(\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,
we will use $\|\cdot\|_q$ to denote $L^q(\mathcal O)$ or $L^q(\mathbb T^2)$ norms of the corresponding function,
if the meaning is clear from the context.

We are now in the position to state the main result of this paper.

We are now in the position to state the main result of this paper.

\begin{proposition}
 \label{PROP-APR}
Given $(v_{0}, \sigma_{0}, p_{0})$ satisfying (\ref{ASSUMIC1})--(\ref{ASSUMIC2}). Let $X_0$ be an arbitrary constant such that
$\|v_{0}\|_{H^3}^2+\|\sigma_{0}\|_{H^2}^2+\|p_0\|_{H^3}^2\leq X_0$. Then, there is a positive time $\mathcal T_0$,
depending only on $\gamma$, $\nu$, $\mu$, $\lambda$, $\underline\sigma$, $\underline p$, and
 $X_0$, such that system (\ref{EQv})--(\ref{EQp}),
 subject to (\ref{BC1})--(\ref{IC}), has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$,
 satisfying
 \begin{eqnarray*}
 &&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\geq0.5\underline\sigma,\quad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\geq0.5\underline p,\\
 &&\sup_{0\leq t\leq\mathcal T_0}(\|v\|_{H^3}^2+\|\sigma\|_{H^2}^2+\|p\|_{H^3}^2)(t)+\epsilon\int_0^{\mathcal T_0}(\|\nabla_h\sigma\|_{H^2}^2+\|\nabla_hp\|_{H^3}^2)dt\nonumber\\
 &&\, \, ~~~+\int_0^{\mathcal T_0}(\|v\|_{H^4}^2+\|\partial_z\sigma\|_{H^2}^2+\|\partial_tv\|_{H^2}^2+\|\partial_t\sigma\|_{H^1}^2+\|\partial_tp\|_{H^2}^2)dt \leq K_0,
 \end{eqnarray*}
 for a positive constant $K_0$ depending only on $\gamma$, $\nu$, $\mu$, $\lambda$, $\underline\sigma$, $\underline p$, and
 $X_0$.
\end{proposition}

By Proposition \ref{PROP-APR}, there is a positive time $\mathcal T_0$ depending only on $\gamma,$ $\nu,$ $\mu$, $\lambda$,
$\underline\sigma$, $\underline p$, and $X_0$, such that for any $\epsilon\in(0,\epsilon_0)$, system \eqref{EQv}--\eqref{EQp}, subject to \eqref{BC1}--\eqref{IC}, has a unique solution $(v_\epsilon,
\sigma_\epsilon, p_\epsilon)$ on $\mathcal O\times(0,\mathcal T_0)$,
with initial data $(v_{0,\epsilon}, \sigma_{0,\epsilon}, p_{0,\epsilon})$, satisfying
 \begin{eqnarray}
 &&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma_\epsilon\geq0.5\underline\sigma,\quad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p_\epsilon\geq0.5\underline p,\label{APRLBsp}\\
 &&\sup_{0\leq t\leq\mathcal T_0}(\|v_\epsilon\|_{H^3}^2+\|\sigma_\epsilon\|_{H^2}^2+\|p_\epsilon\|_{H^3}^2)(t)
 +\epsilon\int_0^{\mathcal T_0}(\|\nabla_h\sigma_\epsilon\|_{H^2}^2+\|\nabla_hp_\epsilon\|_{H^3}^2)dt\nonumber\\
 &&\, \, ~~~+\int_0^{\mathcal T_0}(\|v_\epsilon\|_{H^4}^2+\|\partial_z\sigma_\epsilon\|_{H^2}^2+\|\partial_tv_\epsilon\|_{H^2}^2+\|\partial_t\sigma_\epsilon
 \|_{H^1}^2+\|\partial_tp_\epsilon\|_{H^2}^2)dt \leq K_0,\label{APRHk}
 \end{eqnarray}
 for a positive constant $K_0$ depending only on $\gamma$, $\nu$, $\mu$, $\lambda$, $\underline\sigma$, $\underline p$, and
 $X_0$.

Thanks to \eqref{APRHk}, by the Banach-Alaoglu theorem, and using Cantor's diagonal argument,
 there is a subsequence $(v_{\epsilon_n}, \sigma_{\epsilon_n}, p_{\epsilon_n})$ and a triple $(v,\sigma,p)$,
 such that
 \begin{eqnarray}
 &&v_{\epsilon_n}\stackrel{\ast}{\rightharpoonup}v,\quad\mbox{in }L^\infty(0,\mathcal T_0; H^3(\mathcal O)),\label{WeaC1}\\
 &&\sigma_{\epsilon_n}\stackrel{\ast}{\rightharpoonup}\sigma,\quad\mbox{in }L^\infty(0,\mathcal T_0; H^2(\mathcal O)),\label{WeaC2}\\
 &&p_{\epsilon_n}\stackrel{\ast}{\rightharpoonup}p,\quad\mbox{in }L^\infty(0,\mathcal T_0; H^3(\mathbb T^2)),\label{WeaC3}\\
 &&v_{\epsilon_n}{\rightharpoonup}v,\quad\mbox{in }L^2(0,\mathcal T_0; H^4(\mathcal O)),\label{WeaC4}\\
 &&\partial_z\sigma_{\epsilon_n} {\rightharpoonup}\partial_z\sigma,\quad\mbox{in }L^2(0,\mathcal T_0; H^2(\mathcal O)),\label{WeaC5}\\
 &&\partial_tv_{\epsilon_n} {\rightharpoonup}\partial_tv,\quad\mbox{in }L^2(0,\mathcal T_0; H^2(\mathcal O)),\label{WeaC6}\\
 &&\partial_t\sigma_{\epsilon_n} {\rightharpoonup}\partial_t\sigma,\quad\mbox{in }L^2(0,\mathcal T_0; H^1(\mathcal O)),\label{WeaC7}\\
 &&\partial_tp_{\epsilon_n} {\rightharpoonup}\partial_tp,\quad\mbox{in }L^2(0,\mathcal T_0; H^2(\mathbb T^2)),\label{WeaC8}
 \end{eqnarray}
 where $\stackrel{\ast}{\rightharpoonup}$ and $\rightharpoonup$ represent the weak-* and weak convergence, respectively, in the corresponding spaces. With the aid of the above convergence, noticing that $H^3(\mathcal O)\hookrightarrow\hookrightarrow H^2(\mathcal O)$, $H^2(\mathcal O)\hookrightarrow\hookrightarrow H^1(\mathcal O)\cap C(\overline{\mathcal O})$, and $H^3(\mathbb T^2)\hookrightarrow\hookrightarrow H^2(\mathbb T^2)\cap C(\mathbb T^2)$,
 it follows from the Aubin-Lions lemma that
 \begin{eqnarray}
 &&v_{\epsilon_n} \rightarrow v,\quad\mbox{in }C([0,\mathcal T_0]; H^2(\mathcal O)),\label{StrC1}\\
 &&\sigma_{\epsilon_n} \rightarrow\sigma,\quad\mbox{in }C([0,\mathcal T_0]; H^1(\mathcal O)\cap C(\overline{\mathcal O})),\label{StrC2}\\
 &&\partial_z\sigma_{\epsilon_n} \rightarrow\partial_z\sigma,\quad\mbox{in }L^2(0,\mathcal T_0; H^1(\mathcal O)),\\
 &&p_{\epsilon_n} \rightarrow p,\quad\mbox{in }C([0,\mathcal T_0]; H^2(\mathbb T^2)\cap C(\mathbb T^2)).\label{StrC3}
 \end{eqnarray}
 These imply that $(v,\sigma,p)$ has initial data $(v_0,\sigma_0,p_0)$, enjoys all regularities stated in Theorem \ref{thmmain}, and fulfills the boundary conditions \eqref{BC1} and \eqref{BC2}.
 Besides, by \eqref{StrC2} and \eqref{StrC3}, it follows from \eqref{APRLBsp} that
 \begin{equation*}
 \inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma \geq0.5\underline\sigma,\quad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p \geq0.5\underline p.
 \end{equation*}
 By \eqref{WeaC1}--\eqref{StrC3}, one can take the limit as $n\rightarrow\infty$ to \eqref{EQv}--\eqref{EQp} to show that $(v,\sigma,p)$ satisfies system \eqref{EQv0}--\eqref{EQp0} in the sense of distribution and further pointwisely, by the regularities of $(v,\sigma,p)$. Therefore, $(v,\sigma,p)$ is a strong solution
 to system \eqref{EQv}--\eqref{EQp}, subject to \eqref{BC1}--\eqref{IC}, on $\mathcal O\times(0,\mathcal T_0)$.

\section{Appendix B: local existence of the regularized system}
\label{APPENDIXB}
Denote
\begin{eqnarray*}
 N_1(v,\sigma,p):=-[(v\cdot\nabla_h)v+w\partial_zv+\sigma\nabla_hp],\label{N1}\\
 N_2(v,\sigma,p):=-[v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma(\phi(v,p)-\text{div}_hv)],\label{N2}\\
 N_3(v,p):=-(\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v)+(\gamma-1)\overline{Q(\nabla v)},\label{N3}
\end{eqnarray*}
where $w, \phi(v,p), \bar f,$ and $Q(\nabla v)$ are expressed, respectively, by\eqref{EQw00'}, \eqref{phi}, \eqref{TBf}, and \eqref{Q}.
Recall $\mathcal O=\mathbb T^2\times(0,1)$. Define a solution mapping
\begin{equation}\label{DEFF}
(v,\sigma,p)\mapsto(V,\Sigma,P)=:\mathfrak{F}(v,\sigma,p),
\end{equation}
where $(V,\Sigma,P)$ is the unique solution to the following linear system
\begin{eqnarray}
 \partial_tV-\mu\sigma\Delta V-(\mu+\lambda)\sigma\nabla_h\text{div}_hV&=&N_1(v,\sigma,p),\label{AEQV}\\
 \partial_t\Sigma-\nu\sigma\partial_z^2\Sigma-\epsilon\Delta_h\Sigma&=&N_2(v,\sigma,p),\label{AEQSIGMA}\\
 \partial_tP-\epsilon\Delta_hP&=&N_3(v,p), \label{AEQP}
\end{eqnarray}
in $\mathcal O\times(0,T)$, subject to
\begin{eqnarray}
 &V,\Sigma, P\text{ are periodic in }x,y, \quad\partial_zV|_{z=0,1}=0,\quad \partial_z\Sigma|_{z=0,1}=0,\label{ABC}\\
 &(V,\Sigma,P)|_{t=0}=(v_0,\sigma_0,p_0),\label{AIC}
\end{eqnarray}
where $(v_0,\sigma_0)\in H^3(\mathcal O)$, $p_0\in H^3(\mathbb T^2),$ and
\begin{equation*}
 \partial_zv_0|_{z=0,1}=0,\quad \partial_z\sigma_0|_{z=0,1}=0,\quad\sigma_0\geq\underline\sigma, \quad p_0\geq\underline p,\label{A1}
\end{equation*}
for two positive numbers $\underline\sigma$ and $\underline p$.

Some comments on the proof of Theorem \ref{thmmain} are presented as follows.
Regularity assumption $\sigma_0\in H^2(\mathcal O)$ comes from the following observation: for the toy model
$$
\partial_t\sigma=\nu\sigma\partial_z^2\sigma,
$$
in order that $H^k(\mathcal O)$ regularity propagates, one needs $k\geq2$.
Fortunately, such $H^2(\mathcal O)$ regularity on $\sigma_0$ is still sufficient for the more complicated equation \eqref{EQs0}.
While for the aim of getting $H^2(\mathcal O)$ estimate on $\sigma$, one may apply $\Delta$ to \eqref{EQs0} and, as a result, the
quantity $\Delta\phi(v,p)$ and further, recalling the expression of $\phi(v,p)$, $\nabla_h\Delta_hp$ will be encountered.
In order to get information on $\nabla_h\Delta_hp$, by equation \eqref{EQp0}, one requires $p_0\in H^3(\mathbb T^2)$ and moreover
the quantity $\nabla_h^4v$ will be encountered. Due to this and by \eqref{EQv0}, one has to assume $v_0\in H^3(\mathcal O)$.
The existence of solutions to system \eqref{EQv0}--\eqref{EQp0}, subject to \eqref{BC1}--\eqref{IC}, is established by parabolic
regularization argument. A regularized system is introduced by adding the regularizing dissipative terms $\epsilon\Delta_h\sigma$ and $\epsilon\Delta_hp$
to \eqref{EQs0} and \eqref{EQp0}, respectively, see \eqref{EQv}--\eqref{EQp}, below. Such a regularized system is a semi-linear
parabolic system. The main effort of the proof is then to carry out
suitable $\epsilon$-independent {\it a priori} estimates and the local existence
follows passing to the limit $\epsilon\rightarrow0$.
Continuous dependence on the initial data, which in particular implies
the uniqueness, is proved by performing energy estimates to the difference between two solutions. We employ $L^2$ energy estimate to the
subtracted $\sigma$ equation which contains a term involving $\nabla_hp$. This, due to the absence of dissipation
in the pressure equation, forces us to perform $H^1(\mathbb T^2)$ type energy estimate to the subtracted pressure equation and
further $H^1(\mathcal O)$ energy to the subtracted velocity equations.

The rest of this paper is arranged as follows: in the next section, Section \ref{sec3}, we consider the $\epsilon$-regularized system
and perform suitable a priori estimates; the main result, Theorem \ref{thmmain}, is proved in Section \ref{sec4}; some commutator estimates are proved in Appendix A; local existence and uniqueness of the $\epsilon$-regularized system is proved in Appendix B.

For Theorem~\ref{thmmain}, the authors explain the proof strategy and why the initial regularities are chosen.

- **Why $\sigma_0\in H^2(\mathcal O)$:** From the toy model $\partial_t\sigma=\nu\sigma\partial_z^2\sigma$, “in order that $H^k(\mathcal O)$ regularity propagates, one needs $k\ge 2$.” This $H^2$ assumption is said to be “still sufficient for the more complicated equation \eqref{EQs0}.”

- **Why $p_0\in H^3(\mathbb T^2)$ and $v_0\in H^3(\mathcal O)$:** To get an “$H^2(\mathcal O)$ estimate on $\sigma$,” one applies “$\Delta$ to \eqref{EQs0},” which leads to “$\Delta\phi(v,p)$” and hence to “$\nabla_h\Delta_h p$.” To control $\nabla_h\Delta_h p$ using \eqref{EQp0}, one needs “$p_0\in H^3(\mathbb T^2)$,” and in the process “the quantity $\nabla_h^4 v$ will be encountered,” which (via \eqref{EQv0}) “has to assume $v_0\in H^3(\mathcal O)$.”

- **Existence via parabolic regularization and $\epsilon$-independent estimates:** Existence is established by a “parabolic regularization argument.” They introduce a regularized system by adding “the regularizing dissipative terms $\epsilon\Delta_h\sigma$ and $\epsilon\Delta_h p$ to \eqref{EQs0} and \eqref{EQp0}, respectively.” The resulting system is “a semi-linear parabolic system.” The “main effort” is to derive “suitable $\epsilon$-independent {\it a priori} estimates,” and then “the local existence follows passing to the limit $\epsilon\to 0$.”

- **Uniqueness/continuous dependence by energy estimates on differences:** “Continuous dependence on the initial data, which in particular implies the uniqueness,” is proved by “performing energy estimates to the difference between two solutions.” Specifically, they use an “$L^2$ energy estimate to the subtracted $\sigma$ equation” but this equation has “a term involving $\nabla_h p$.” Because of “the absence of dissipation in the pressure equation,” they are “force[d]” to do an “$H^1(\mathbb T^2)$ type energy estimate to the subtracted pressure equation” and “further $H^1(\mathcal O)$ energy to the subtracted velocity equations.”

For the main theorem, the authors explain the proof strategy and why the initial regularities are chosen.

- **Why $\sigma_0\in H^2(\mathcal O)$:** From the toy model $\partial_t\sigma=\nu\sigma\partial_z^2\sigma$, “in order that $H^k(\mathcal O)$ regularity propagates, one needs $k\ge 2$.” This $H^2$ assumption is said to be “still sufficient for the more complicated equation
\begin{eqnarray}
  &\partial_tv+(v\cdot\nabla_h)v+w\partial_zv+\sigma\nabla_hp=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\text{div}_hv,\label{EQv0}\\
  &w=\nu\partial_z\sigma-\int_0^z\phi(v,p)dz',\label{EQw0}\\
  &\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma(\phi(v,p)-\text{div}_hv)=\nu\sigma\partial_z^2\sigma,
  \label{EQs0}\\
  &\partial_tp+\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)}, \label{EQp0}
\end{eqnarray}
.”

- **Why $p_0\in H^3(\mathbb T^2)$ and $v_0\in H^3(\mathcal O)$:** To get an “$H^2(\mathcal O)$ estimate on $\sigma$,” one applies “$\Delta$ to” the $\sigma$-equation in
\begin{eqnarray}
  &\partial_tv+(v\cdot\nabla_h)v+w\partial_zv+\sigma\nabla_hp=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\text{div}_hv,\label{EQv0}\\
  &w=\nu\partial_z\sigma-\int_0^z\phi(v,p)dz',\label{EQw0}\\
  &\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma(\phi(v,p)-\text{div}_hv)=\nu\sigma\partial_z^2\sigma,
  \label{EQs0}\\
  &\partial_tp+\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)}, \label{EQp0}
\end{eqnarray}
which leads to “$\Delta\phi(v,p)$” and hence to “$\nabla_h\Delta_h p$.” To control $\nabla_h\Delta_h p$ using the pressure equation in the same system (i.e. the $p$-equation displayed above), one needs “$p_0\in H^3(\mathbb T^2)$,” and in the process “the quantity $\nabla_h^4 v$ will be encountered,” which (via the velocity equation in the displayed system above) “has to assume $v_0\in H^3(\mathcal O)$.”

- **Existence via parabolic regularization and $\epsilon$-independent estimates:** Existence is established by a “parabolic regularization argument.” They introduce a regularized system by adding “the regularizing dissipative terms $\epsilon\Delta_h\sigma$ and $\epsilon\Delta_h p$” to the $\sigma$- and $p$-equations in
\begin{eqnarray}
  &\partial_tv+(v\cdot\nabla_h)v+w\partial_zv+\sigma\nabla_hp=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\text{div}_hv,\label{EQv0}\\
  &w=\nu\partial_z\sigma-\int_0^z\phi(v,p)dz',\label{EQw0}\\
  &\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma(\phi(v,p)-\text{div}_hv)=\nu\sigma\partial_z^2\sigma,
  \label{EQs0}\\
  &\partial_tp+\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)}, \label{EQp0}
\end{eqnarray}
respectively. The resulting system is “a semi-linear parabolic system.” The “main effort” is to derive “suitable $\epsilon$-independent {\it a priori} estimates,” and then “the local existence follows passing to the limit $\epsilon\to 0$.”

- **Uniqueness/continuous dependence by energy estimates on differences:** “Continuous dependence on the initial data, which in particular implies the uniqueness,” is proved by “performing energy estimates to the difference between two solutions.” Specifically, they use an “$L^2$ energy estimate to the subtracted $\sigma$ equation” but this equation has “a term involving $\nabla_h p$.” Because of “the absence of dissipation in the pressure equation” (namely, in the $p$-equation in the displayed system above), they are “force[d]” to do an “$H^1(\mathbb T^2)$ type energy estimate to the subtracted pressure equation” and “further $H^1(\mathcal O)$ energy to the subtracted velocity equations.”

\label{thmmain}
Given $v_{0}\in H^3(\mathcal O)$, $\sigma_{0} \in H^2(\mathcal O)$, and $p_{0}\in H^3(\mathbb T^2),$ such that
$$
\sigma_{0}\geq\underline\sigma, \quad p_{0}\geq\underline p,\quad \partial_zv_{0}|_{z=0,1}=0,\quad \partial_z\sigma_{0}|_{z=0,1}=0,
$$
for two positive numbers $\underline\sigma$ and $\underline p$.
Then, there is a positive time $\mathcal T_0$,
depending only on $\gamma$, $\nu$, $\mu$, $\lambda$, $\underline\sigma$, $\underline p$, and
  $\|v_{0}\|_{H^3}^2+\|\sigma_{0}\|_{H^2}^2+\|p_0\|_{H^3}^2$, such that the system
\begin{eqnarray}
  &\partial_tv+(v\cdot\nabla_h)v+w\partial_zv+\sigma\nabla_hp=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\text{div}_hv,\label{EQv0}\\
  &w=\nu\partial_z\sigma-\int_0^z\phi(v,p)dz',\label{EQw0}\\
  &\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma(\phi(v,p)-\text{div}_hv)=\nu\sigma\partial_z^2\sigma,
  \label{EQs0}\\
  &\partial_tp+\bar v\cdot\nabla_hp+\gamma p\text{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)}, \label{EQp0}
\end{eqnarray}
  subject to
\begin{eqnarray}
  &v, \sigma, \mbox{ and } p \mbox{ are periodic in }x, y, \label{BC1}\\
  &\partial_zv|_{z=0,1}=0, \quad\partial_z\sigma|_{z=0,1}=0. \label{BC2}
\end{eqnarray}
and
\begin{equation}
  \label{IC}
  (v, \sigma, p)|_{t=0}=(v_{0}, \sigma_{0}, p_{0}).
\end{equation}
has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$,
  satisfying
  \begin{eqnarray*}
  &&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\geq0.5\underline\sigma,\quad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\geq0.5\underline p,\\
    &&v\in C([0,\mathcal T_0]; H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0; H^3(\mathcal O))\cap L^2(0,\mathcal T_0; H^4(\mathcal O)),\\
    &&\sigma\in C([0,\mathcal T_0]; H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0; H^2(\mathcal O)),\quad \partial_z\sigma\in L^2(0,\mathcal T_0; H^2(\mathcal O)),\\
    &&\partial_tv\in L^2(0,\mathcal T_0; H^2(\mathcal O)),\quad \partial_t\sigma\in L^2(0,\mathcal T_0; H^1(\mathcal O)),\\
    &&p\in C([0,\mathcal T_0]; H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0; H^3(\mathbb T^2)),\quad\partial_tp\in  L^2(0,\mathcal T_0; H^2(\mathbb T^2)).
  \end{eqnarray*},

Question:
Let $\mathcal O=\mathbb T^2\times(0,1)$. For a function $f(x,y,z)$, define
$$\bar f(x,y):=\int_0^1 f(x,y,z)\,dz,\qquad \tilde f:=f-\bar f.$$
Let
$$Q(\nabla v):=\bigl[\mu(\nabla_h v+\nabla_h^{\top}v)+\lambda\,\mathrm{div}_h v\,\mathbb I_2\bigr]:\nabla_h v+\mu|\partial_z v|^2,$$
$$\phi(v,p):=\mathrm{div}_h\tilde v+\frac{1}{\gamma p}\Bigl(\tilde v\cdot\nabla_h p-(\gamma-1)\widetilde{Q(\nabla v)}\Bigr),$$
$$w=\nu\partial_z\sigma-\int_0^z \phi(v,p)(x,y,z',t)\,dz'.$$
Assume
$$v_0\in H^3(\mathcal O),\qquad \sigma_0\in H^2(\mathcal O),\qquad p_0\in H^3(\mathbb T^2),$$
and for some positive constants $\underline\sigma,\underline p$,
$$\sigma_0\ge \underline\sigma,\qquad p_0\ge \underline p,\qquad \partial_z v_0|_{z=0,1}=0,\qquad \partial_z\sigma_0|_{z=0,1}=0.$$
Which statement holds for the problem of finding $v,\sigma$ on $\mathcal O\times(0,T)$ and $p$ on $\mathbb T^2\times(0,T)$ such that
\begin{eqnarray*}
&&\partial_t v+(v\cdot\nabla_h)v+w\partial_z v+\sigma\nabla_h p=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\mathrm{div}_h v,\\
&&\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma\bigl(\phi(v,p)-\mathrm{div}_h v\bigr)=\nu\sigma\partial_z^2\sigma,\\
&&\partial_t p+\bar v\cdot\nabla_h p+\gamma p\,\mathrm{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)},
\end{eqnarray*}
with $v,\sigma,p$ periodic in $(x,y)$, boundary conditions $\partial_z v|_{z=0,1}=0$ and $\partial_z\sigma|_{z=0,1}=0$, and initial data $(v,\sigma,p)|_{t=0}=(v_0,\sigma_0,p_0)$?

There exists a time $\mathcal T_0>0$, depending only on $\gamma,\nu,\mu,\lambda,\underline\sigma,\underline p$, and $\|v_0\|_{H^3}^2+\|\sigma_0\|_{H^2}^2+\|p_0\|_{H^3}^2$, such that the above system has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$ satisfying
\begin{eqnarray*}
&&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,\qquad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,\\
&&v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),\\
&&\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\\
&&\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),\\
&&p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\end{eqnarray*}

Choices:
B.
There exists a time $\mathcal T_0>0$, depending only on $\gamma,\nu,\mu,\lambda,\underline\sigma,\underline p$, and $\|v_0\|_{H^3}^2+\|\sigma_0\|_{H^1}^2+\|p_0\|_{H^2}^2$, such that the above system has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$ satisfying
\begin{eqnarray*}
&&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,\qquad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,\\
&&v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),\\
&&\sigma\in C([0,\mathcal T_0];L^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^1(\mathcal O)),\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),\\
&&\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;L^2(\mathcal O)),\\
&&p\in C([0,\mathcal T_0];H^1(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^2(\mathbb T^2)),\qquad \partial_t p\in L^2(0,\mathcal T_0;H^1(\mathbb T^2)).
\end{eqnarray*}
C.
There exists a time $\mathcal T_0>0$, depending only on $\gamma,\nu,\mu,\lambda,\underline\sigma,\underline p$, and $\|v_0\|_{H^3}^2+\|\sigma_0\|_{H^2}^2+\|p_0\|_{H^3}^2$, such that the above system has at least one local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$ satisfying
\begin{eqnarray*}
&&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,\qquad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,\\
&&v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),\\
&&\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\\
&&\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),\\
&&p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\end{eqnarray*}
D.
There exists a universal time $\mathcal T_0>0$, depending only on $\gamma,\nu,\mu,\lambda,\underline\sigma,$ and $\underline p$ but independent of $\|v_0\|_{H^3},\|\sigma_0\|_{H^2},\|p_0\|_{H^3}$, such that the above system has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$ satisfying
\begin{eqnarray*}
&&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,\qquad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,\\
&&v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),\\
&&\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\\
&&\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),\\
&&p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\end{eqnarray*}
E.
There exists a time $\mathcal T_0>0$, depending only on $\gamma,\nu,\mu,\lambda,\underline\sigma,\underline p$, and $\|v_0\|_{H^3}^2+\|\sigma_0\|_{H^2}^2+\|p_0\|_{H^3}^2$, such that the above system has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$ satisfying
\begin{eqnarray*}
&&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\ge \underline\sigma,\qquad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\ge \underline p,\\
&&v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),\\
&&\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\\
&&\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),\\
&&p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\end{eqnarray*}

minimal initial regularity for $\sigma_0$ and $p_0$

dropped uniqueness while keeping the same existence/regularity conclusion

dependence of lifespan on Sobolev norms of initial data

strict preservation of the full lower bounds instead of the halved bounds

The stem does not explicitly state the correct existence theorem. It presents the PDE system and assumptions, but the reader must still distinguish among uniqueness, regularity, and lifespan-dependence claims.

This is close to a theorem-identification item: the task is essentially to pick the correct local well-posedness statement for the displayed system. The alternatives introduce some variation, but the question still largely tests recognition of the theorem rather than a genuinely independent conclusion.

Moderate reasoning is needed to assess which statement is most plausible, especially regarding uniqueness, top-order time continuity, and dependence on positivity lower bounds. However, it does not strongly force derivation or synthesis beyond theorem-level matching.

The distractors are mathematically plausible and target realistic failure modes: weakened initial regularity, loss of uniqueness, incorrect parameter dependence, and an overly strong regularity upgrade. They are distinct and well aligned with common PDE theorem confusions.

A solid but theorem-recognition-heavy MCQ: little answer leakage and strong distractors, but only moderate generative reasoning and limited avoidance of theorem-restatement structure.


Question:
Let $\mathcal O=\mathbb T^2\times(0,1)$. For a function $f(x,y,z)$, define
$$\bar f(x,y):=\int_0^1 f(x,y,z)\,dz,\qquad \tilde f:=f-\bar f.$$
Let
$$Q(\nabla v):=\bigl[\mu(\nabla_h v+\nabla_h^{\top}v)+\lambda\,\mathrm{div}_h v\,\mathbb I_2\bigr]:\nabla_h v+\mu|\partial_z v|^2,$$
$$\phi(v,p):=\mathrm{div}_h\tilde v+\frac{1}{\gamma p}\Bigl(\tilde v\cdot\nabla_h p-(\gamma-1)\widetilde{Q(\nabla v)}\Bigr),$$
$$w=\nu\partial_z\sigma-\int_0^z \phi(v,p)(x,y,z',t)\,dz'.$$
Assume
$$v_0\in H^3(\mathcal O),\qquad \sigma_0\in H^2(\mathcal O),\qquad p_0\in H^3(\mathbb T^2),$$
and for some positive constants $\underline\sigma,\underline p$,
$$\sigma_0\ge \underline\sigma,\qquad p_0\ge \underline p,\qquad \partial_z v_0|_{z=0,1}=0,\qquad \partial_z\sigma_0|_{z=0,1}=0.$$
Which statement holds for the problem of finding $v,\sigma$ on $\mathcal O\times(0,T)$ and $p$ on $\mathbb T^2\times(0,T)$ such that
\begin{eqnarray*}
&&\partial_t v+(v\cdot\nabla_h)v+w\partial_z v+\sigma\nabla_h p=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\mathrm{div}_h v,\\
&&\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma\bigl(\phi(v,p)-\mathrm{div}_h v\bigr)=\nu\sigma\partial_z^2\sigma,\\
&&\partial_t p+\bar v\cdot\nabla_h p+\gamma p\,\mathrm{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)},
\end{eqnarray*}
with $v,\sigma,p$ periodic in $(x,y)$, boundary conditions $\partial_z v|_{z=0,1}=0$ and $\partial_z\sigma|_{z=0,1}=0$, and initial data $(v,\sigma,p)|_{t=0}=(v_0,\sigma_0,p_0)$?

Choices:
B.
There exists a time $\mathcal T_0>0$, depending only on $\gamma,\nu,\mu,\lambda,\underline\sigma,\underline p$, and $\|v_0\|_{H^3}^2+\|\sigma_0\|_{H^1}^2+\|p_0\|_{H^2}^2$, such that the above system has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$ satisfying
\begin{eqnarray*}
&&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,\qquad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,\\
&&v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),\\
&&\sigma\in C([0,\mathcal T_0];L^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^1(\mathcal O)),\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),\\
&&\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;L^2(\mathcal O)),\\
&&p\in C([0,\mathcal T_0];H^1(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^2(\mathbb T^2)),\qquad \partial_t p\in L^2(0,\mathcal T_0;H^1(\mathbb T^2)).
\end{eqnarray*}
C.
There exists a time $\mathcal T_0>0$, depending only on $\gamma,\nu,\mu,\lambda,\underline\sigma,\underline p$, and $\|v_0\|_{H^3}^2+\|\sigma_0\|_{H^2}^2+\|p_0\|_{H^3}^2$, such that the above system has at least one local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$ satisfying
\begin{eqnarray*}
&&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,\qquad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,\\
&&v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),\\
&&\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\\
&&\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),\\
&&p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\end{eqnarray*}
D.
There exists a universal time $\mathcal T_0>0$, depending only on $\gamma,\nu,\mu,\lambda,\underline\sigma,$ and $\underline p$ but independent of $\|v_0\|_{H^3},\|\sigma_0\|_{H^2},\|p_0\|_{H^3}$, such that the above system has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$ satisfying
\begin{eqnarray*}
&&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,\qquad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,\\
&&v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),\\
&&\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\\
&&\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),\\
&&p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\end{eqnarray*}
E.
There exists a time $\mathcal T_0>0$, depending only on $\gamma,\nu,\mu,\lambda,\underline\sigma,\underline p$, and $\|v_0\|_{H^3}^2+\|\sigma_0\|_{H^2}^2+\|p_0\|_{H^3}^2$, such that the above system has a unique local solution $(v,\sigma,p)$ on $\mathcal O\times(0,\mathcal T_0)$ satisfying
\begin{eqnarray*}
&&\inf_{(x,y,z,t)\in\mathcal O\times[0,\mathcal T_0]}\sigma\ge \underline\sigma,\qquad \inf_{(x,y,t)\in\mathbb T^2\times[0,\mathcal T_0]}p\ge \underline p,\\
&&v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),\\
&&\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\\
&&\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),\\
&&p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\end{eqnarray*}

The stem does not explicitly reveal the correct option. It sets up the PDE system and assumptions, but the exact conclusion must still be selected from several close variants.

This is essentially a direct recognition task for the precise local well-posedness theorem statement. The correct answer is the theorem itself, only contrasted against slightly modified versions.

There is some reasoning pressure because the options differ in subtle but meaningful ways: uniqueness vs. existence, lifespan dependence, lower-bound propagation, and Sobolev regularity. However, the task is still mostly theorem recall/verification rather than generative mathematical reasoning.

The distractors are plausible and mathematically meaningful. They target common failure modes: weakened regularity assumptions, dropping uniqueness, incorrect lifespan dependence, and overstating positivity preservation.

A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely a restatement of a known result and only moderately tests reasoning.