\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


Question:
Let \(\mathcal O=\mathbb T^2\times(0,1)\). Consider unknowns \(v=v(x,y,z,t)\in\mathbb R^2\), \(\sigma=\sigma(x,y,z,t)\), and \(p=p(x,y,t)\), where for any \(f\) we write
\[
\bar f(x,y,t)=\int_0^1 f(x,y,z,t)\,dz,\qquad \tilde f=f-\bar f.
\]
Define
\[
\mathbb S_h=\mu(\nabla_h v+\nabla_h^{\top}v)+\lambda (\operatorname{div}_h v)\mathbb I_2,
\]
\[
Q(\nabla v)=\mathbb S_h:\nabla_h v+\mu|\partial_z v|^2,
\]
\[
\phi(v,p)=\operatorname{div}_h\tilde v+\frac{1}{\gamma p}\Bigl(\tilde v\cdot\nabla_h p-(\gamma-1)\widetilde{Q(\nabla v)}\Bigr),
\]
with \(\widetilde{Q(\nabla v)}=Q(\nabla v)-\overline{Q(\nabla v)}\), and
\[
w=\nu\partial_z\sigma-\int_0^z \phi(v,p)\,dz'.
\]
Assume initial data satisfy
\[
v_0\in H^3(\mathcal O),\qquad \sigma_0\in H^2(\mathcal O),\qquad p_0\in H^3(\mathbb T^2),
\]
\[
\sigma_0\ge \underline\sigma,\qquad p_0\ge \underline p,\qquad \partial_z v_0|_{z=0,1}=0,\qquad \partial_z\sigma_0|_{z=0,1}=0,
\]
for some positive numbers \(\underline\sigma\) and \(\underline p\). Consider the system
\[
\partial_t v+(v\cdot\nabla_h)v+w\partial_z v+\sigma\nabla_h p=\mu\sigma\Delta v+(\mu+\lambda)\sigma\nabla_h\operatorname{div}_h v,
\]
\[
\partial_t\sigma+v\cdot\nabla_h\sigma+w\partial_z\sigma+\sigma\bigl(\phi(v,p)-\operatorname{div}_h v\bigr)=\nu\sigma\partial_z^2\sigma,
\]
\[
\partial_t p+\bar v\cdot\nabla_h p+\gamma p\,\operatorname{div}_h\bar v=(\gamma-1)\overline{Q(\nabla v)},
\]
subject to periodicity of \(v,\sigma,p\) in \((x,y)\), boundary conditions
\[
\partial_z v|_{z=0,1}=0,\qquad \partial_z\sigma|_{z=0,1}=0,
\]
and initial condition
\[
(v,\sigma,p)|_{t=0}=(v_0,\sigma_0,p_0).
\]
Which existence statement holds for this problem?

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 initial-boundary value problem has a unique local solution \((v,\sigma,p)\) on \(\mathcal O\times(0,\mathcal T_0)\) satisfying
\[
\inf_{(x,y,z,t)\in \mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,
\qquad
\inf_{(x,y,t)\in \mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,
\]
\[
v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),
\]
\[
\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),
\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),
\]
\[
\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),
\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),
\]
\[
p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),
\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\]

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 initial-boundary value problem has a unique local solution \((v,\sigma,p)\) on \(\mathcal O\times(0,\mathcal T_0)\) satisfying
\[
\inf_{(x,y,z,t)\in \mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,
\qquad
\inf_{(x,y,t)\in \mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,
\]
\[
v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),
\]
\[
\sigma\in C([0,\mathcal T_0];L^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^1(\mathcal O)),
\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),
\]
\[
\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),
\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;L^2(\mathcal O)),
\]
\[
p\in C([0,\mathcal T_0];H^1(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^2(\mathbb T^2)),
\qquad \partial_t p\in L^2(0,\mathcal T_0;H^1(\mathbb T^2)).
\]
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 initial-boundary value problem has at least one local solution \((v,\sigma,p)\) on \(\mathcal O\times(0,\mathcal T_0)\) satisfying
\[
\inf_{(x,y,z,t)\in \mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,
\qquad
\inf_{(x,y,t)\in \mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,
\]
\[
v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),
\]
\[
\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),
\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),
\]
\[
\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),
\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),
\]
\[
p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),
\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\]
D.
There exists a time \(\mathcal T_0>0\), depending only on \(\gamma,\nu,\mu,\lambda\) and the norms \(\|v_0\|_{H^3}^2+\|\sigma_0\|_{H^2}^2+\|p_0\|_{H^3}^2\), but independent of the lower bounds \(\underline\sigma\) and \(\underline p\), such that the above initial-boundary value problem has a unique local solution \((v,\sigma,p)\) on \(\mathcal O\times(0,\mathcal T_0)\) satisfying
\[
\inf_{(x,y,z,t)\in \mathcal O\times[0,\mathcal T_0]}\sigma\ge 0.5\,\underline\sigma,
\qquad
\inf_{(x,y,t)\in \mathbb T^2\times[0,\mathcal T_0]}p\ge 0.5\,\underline p,
\]
\[
v\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),
\]
\[
\sigma\in C([0,\mathcal T_0];H^1(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),
\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),
\]
\[
\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),
\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),
\]
\[
p\in C([0,\mathcal T_0];H^2(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),
\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\]
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 initial-boundary value problem has a unique local solution \((v,\sigma,p)\) on \(\mathcal O\times(0,\mathcal T_0)\) satisfying the same positivity bounds and, moreover,
\[
v\in C([0,\mathcal T_0];H^3(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^3(\mathcal O))\cap L^2(0,\mathcal T_0;H^4(\mathcal O)),
\]
\[
\sigma\in C([0,\mathcal T_0];H^2(\mathcal O))\cap L^\infty(0,\mathcal T_0;H^2(\mathcal O)),
\qquad \partial_z\sigma\in L^2(0,\mathcal T_0;H^2(\mathcal O)),
\]
\[
\partial_t v\in L^2(0,\mathcal T_0;H^2(\mathcal O)),
\qquad \partial_t\sigma\in L^2(0,\mathcal T_0;H^1(\mathcal O)),
\]
\[
p\in C([0,\mathcal T_0];H^3(\mathbb T^2))\cap L^\infty(0,\mathcal T_0;H^3(\mathbb T^2)),
\qquad \partial_t p\in L^2(0,\mathcal T_0;H^2(\mathbb T^2)).
\]

minimal initial regularity for \sigma_0 and p_0

dependence of lifespan on positivity lower bounds

time-continuity upgraded to top-order spaces despite anisotropic diffusion and no pressure dissipation