On weak-strong uniqueness and singular limit for the compressible Primitive Equations

TL;DR

This paper proves that weak solutions to the compressible Primitive Equations are unique when strong solutions exist and demonstrates the convergence of compressible to incompressible inviscid Primitive Equations in low Mach and high Reynolds number regimes.

Contribution

It establishes the weak-strong uniqueness property and the singular limit from compressible to incompressible Primitive Equations, bridging a gap in the theoretical understanding.

Findings

Weak-strong uniqueness of solutions established.

Convergence from compressible to incompressible Primitive Equations proved.

First demonstration of the link between compressible and incompressible inviscid PE.

Abstract

This paper addresses the weak-strong uniqueness property and singular limit for the compressible Primitive Equations (PE). We show that a weak solution coincides with the strong solution emanating from the same initial data. On the other hand, we prove compressible PE will approach the incompressible inviscid PE equations in the regime of low Mach number and large Reynolds number in the case of well-prepared initial data. To the best of the authors' knowledge, this is the first work to bridge the link between the compressible PE with incompressible inviscid PE.

Equations

$$\displaystyle\left\{\begin{array}[]{llll}\partial_{t}\rho+\text{div}_{x}(\rho\mathbf{u})+\partial_{z}(\rho w)=0,\\ \partial_{t}(\rho\mathbf{u})+\textrm{div}_{x}(\rho\mathbf{u}\otimes\mathbf{u})+\partial_{z}(\rho\mathbf{u}w)+\nabla_{x}p(\rho)=\mu\Delta_{x}\mathbf{u}+\lambda\partial^{2}_{zz}\mathbf{u},\\ \partial_{z}p(\rho)=0,\end{array}\right.$$

$$w{|}_{z=0}=w{|}_{z=1}=0,{\partial }_z\mathbf{u}{|}_{z=0}={\partial }_z\mathbf{u}{|}_{z=1}=0,$$

$$\rho \mathbf{u}{|}_{t=0}={\mathbf{m}}_0(x,z),\rho {|}_{t=0}={\rho }_0(x).$$

$$p(\rho )={\rho }^{\gamma }(\gamma >1).$$

$$\mathbf{u}\in L^2(0,T;H^1(\Omega )),\rho |\mathbf{u}{|}^2\in L^{\infty }(0,T;L^1(\Omega )),\rho \in L^{\infty }(0,T;L^{\gamma }(\Omega )\cap L^1(\Omega )).$$

$$[{\int }_{\Omega }\rho \phi dxdz{]}_{t=0}^{t=\tau }={\int }_0^{\tau }{\int }_{\Omega }\rho {\partial }_t\phi +\rho \mathbf{u}{\nabla }_x\phi +\rho w{\partial }_z\phi dxdzdt,$$

$$[{\int }_{\Omega }\rho \mathbf{u}\phi dxdz{]}_{t=0}^{t=\tau }$$

$$-{\int }_0^{\tau }{\int }_{\Omega }[\mu {\nabla }_x\mathbf{u}:{\nabla }_x\phi +\lambda {\partial }_z\mathbf{u}{\partial }_z\phi ]dxdzdt,$$

$$[{\int }_{\Omega }\frac{1}{2}\rho |\mathbf{u}{|}^2+P(\rho )-P^{\prime }(\overline{\rho })(\rho -\overline{\rho })-P(\overline{\rho })dxdz]{|}_{t=0}^{t=\tau }+{\int }_0^{\tau }{\int }_{\Omega }(\mu |{\nabla }_x\mathbf{u}{|}^2+\lambda |{\partial }_z\mathbf{u}{|}^2)dxdzdt\le 0,$$

$$\rho w(x,z,t)=-{\mathrm{div}}_{\mathrm{x}}(\rho \tilde{\mathbf{u}})+\mathrm{z}{\mathrm{div}}_{\mathrm{x}}(\rho \overline{\mathbf{u}}),\text{in the sense of}{\mathrm{H}}^{-1}(\mathrm{\Omega }),$$

$$\tilde{\mathbf{u}}(x,z,t)={\int }_0^{z}u(x,s,t)ds,\overline{u}(x,t)={\int }_0^{1}u(x,z,t)dz.$$

$$\mathcal{E}(\rho ,\mathbf{u}|r,\mathbf{U})$$

$$={\int }_{\Omega }(\frac{1}{2}{\rho }^2|\mathbf{u}|+P(\rho ))dxdz-{\int }_{\Omega }\rho \mathbf{u}\cdot \mathbf{U}dxdz+{\int }_{\Omega }\rho [\frac{|\mathbf{U}{|}^2}{2}-P^{\prime }(r)]dxdz+{\int }_{\Omega }p(r)dxdz$$

$$=\sum _{i=1}^4{I}_i,$$

$$\mathcal{E}$$

$$\le {\int }_0^{\tau }{\int }_{\Omega }\rho (\mathbf{U}-\mathbf{u}){\partial }_t\mathbf{U}+\rho \mathbf{u}(\mathbf{U}-\mathbf{u})\cdot {\nabla }_x\mathbf{U}+\rho w(\mathbf{U}-\mathbf{u})\cdot {\partial }_z\mathbf{U}-p(\rho ){\text{div}}_x\mathbf{U}dxdzdt$$

$$-{\int }_0^{\tau }{\int }_{\Omega }{P}^{\prime \prime }(r)(\rho {\partial }_{t}r+\rho \mathbf{u}{\nabla }_{x}r)dxdzdt+{\int }_0^{\tau }{\int }_{\Omega }{\partial }_{t}p(r)dxdzdt.$$

$$I_1{|}_{t=0}^{t=\tau }+{\int }_0^{\tau }{\int }_{\Omega }(\mu |{\nabla }_x\mathbf{u}{|}^2+\lambda |{\partial }_z\mathbf{u}{|}^2)dxdzdt\le 0,$$

$$I_2{|}_{t=0}^{t=\tau }=-{\int }_0^{\tau }{\int }_{\Omega }\rho \mathbf{u}{\partial }_t\mathbf{U}+\rho \mathbf{u}\otimes \mathbf{u}:{\nabla }_x\mathbf{U}+\rho \mathbf{u}w{\partial }_z\mathbf{U}+p(\rho ){\text{div}}_x\mathbf{U}dxdzdt$$

$$+{\int }_0^{\tau }{\int }_{\Omega }\mu {\nabla }_x\mathbf{u}:{\nabla }_x\mathbf{U}+\lambda {\partial }_z\mathbf{u}{\partial }_z\mathbf{U}dxdzdt,$$

$$I_3{|}_{t=0}^{t=\tau }={\int }_0^{\tau }{\int }_{\Omega }\rho {\partial }_t\frac{|\mathbf{U}{|}^2}{2}+\rho \mathbf{u}\cdot {\nabla }_x\frac{|\mathbf{U}{|}^2}{2}+\rho w{\partial }_z\frac{|\mathbf{U}{|}^2}{2}dxdzdt$$

$$-{\int }_0^{\tau }{\int }_{\Omega }\rho {\partial }_t{P}^{\prime }(r)+\rho \mathbf{u}\cdot {\nabla }_x{P}^{\prime }(r)+\rho w{\partial }_z{P}^{\prime }(r)dxdzdt$$

$$={\int }_0^{\tau }{\int }_{\Omega }\rho \mathbf{U}{\partial }_t\mathbf{U}+\rho \mathbf{uU}\cdot {\nabla }_x\mathbf{U}+\rho w\mathbf{U}{\partial }_z\mathbf{U}dxdzdt$$

$$-{\int }_0^{\tau }{\int }_{\Omega }\rho P^{\prime \prime }(r){\partial }_{t}r+P^{\prime \prime }(r)\rho \mathbf{u}\cdot {\nabla }_{x}rdxdzdt,$$

$$I_4{|}_{t=0}^{t=\tau }=[{\int }_{\Omega }p(\rho )dxdz]{|}_{t=0}^{t=\tau }.$$

$$\mathcal{E}$$

$$\le {\int }_0^{\tau }{\int }_{\Omega }\rho (\mathbf{U}-\mathbf{u}){\partial }_t\mathbf{U}+\rho \mathbf{u}(\mathbf{U}-\mathbf{u})\cdot {\nabla }_x\mathbf{U}+\rho w(\mathbf{U}-\mathbf{u})\cdot {\partial }_z\mathbf{U}-p(\rho ){\text{div}}_x\mathbf{U}dxdzdt$$

$$-{\int }_0^{\tau }{\int }_{\Omega }{P}^{\prime \prime }(r)(\rho {\partial }_{t}r+\rho \mathbf{u}{\nabla }_{x}r)dxdzdt+{\int }_0^{\tau }{\int }_{\Omega }{\partial }_{t}p(r)dxdzdt.$$

$$r^{\frac{1}{2}}\in L^{\infty }(0,T;H^2(\Omega )),{\partial }_t{r}^{\frac{1}{2}}\in L^{\infty }(0,T;H^1(\Omega )),r>0\text{for all}(t,x),$$

$$\mathbf{U}\in L^{\infty }(0,T;H^3(\Omega ))\cap L^2(0,T;H^4(\Omega )),{\partial }_t\mathbf{U}\in L^2(0,T;H^2(\Omega )),$$

$$\rho =r,\mathbf{u}=\mathbf{U},\text{in}(0,T)\times \Omega .$$

$${\int }_{\Omega }\rho \mathbf{u}(\mathbf{U}-\mathbf{u})\cdot {\nabla }_x\mathbf{U}dxdz={\int }_{\Omega }\rho (\mathbf{u}-\mathbf{U})(\mathbf{U}-\mathbf{u})\cdot {\nabla }_x\mathbf{U}dxdz+{\int }_{\Omega }\rho \mathbf{U}(\mathbf{U}-\mathbf{u})\cdot {\nabla }_x\mathbf{U}dxdz.$$

$${\int }_{\Omega }\rho (\mathbf{u}-\mathbf{U})(\mathbf{U}-\mathbf{u})\cdot {\nabla }_x\mathbf{U}dxdz\le C\mathcal{E}(\rho ,\mathbf{u}|r,\mathbf{U}).$$

$$(r\mathbf{U}{)}_t+\text{div}(r\mathbf{U}\otimes \mathbf{U})+{\partial }_z(r\mathbf{U}W)+{\nabla }_{x}p(r)=\mu {\Delta }_x\mathbf{U}+\lambda {\partial }_{zz}\mathbf{U},$$

$${\mathbf{U}}_t+\mathbf{U}\cdot {\nabla }_x\mathbf{U}+W{\partial }_z\mathbf{U}=-\frac{1}{r}{\nabla }_{x}p(r)+\frac{\mu }{r}{\Delta }_x\mathbf{U}+\frac{\lambda }{r}{\partial }_{zz}\mathbf{U}.$$

$${\int }_{\Omega }\rho (\mathbf{U}-\mathbf{u})\cdot {\partial }_t\mathbf{U}+\rho \mathbf{U}(\mathbf{U}-\mathbf{u})\cdot {\nabla }_x\mathbf{U}+\rho W(\mathbf{U}-\mathbf{u})\cdot {\partial }_z\mathbf{U}+\rho (w-W)(\mathbf{U}-\mathbf{u})\cdot {\partial }_z\mathbf{U}dxdz$$

$$={\int }_{\Omega }\frac{\rho }{r}(\mathbf{U}-\mathbf{u})(-{\nabla }_{x}p(r)+\mu {\Delta }_x\mathbf{U}+\lambda {\partial }_{zz}\mathbf{U})dxdz+{\int }_{\Omega }\rho (w-W)(\mathbf{U}-\mathbf{u})\cdot {\partial }_z\mathbf{U}dxdz.$$

$$\mathcal{E}$$

Full text

Key words: compressible Primitive Equations, singular limit, low Mach number, weak-strong uniqueness.

2010 Mathematics Subject Classifications: 35Q30.

1 Introduction

The earth is surrounded and occupied by atmosphere and ocean, which play an important role in human’s life. From the mathematical point of view and numerical perspective, it is very complicated to use the full hydrodynamical and thermodynamical equations to describe the motion and fascinating phenomena of atmosphere and ocean. In order to simplify model, scientists introduce the Primitive Equations (PE) model in meteorology and geophysical fluid dynamics, which helps us to predict the long-term weather and detect the global climate changes. In this paper, we study the following Compressible Primitive Equations (CPE):

[TABLE]

in $(0,T)\times\Omega$. Here $\Omega=\{(x,z)|x\in\mathbb{T}^{2},0<z<1\}$, $x$ denotes the horizontal direction and $z$ denotes the vertical direction. $\rho=\rho(t,x)$, $\mathbf{u}(t,x,z)\in\mathbb{R}^{2}$ and $w(t,x,z)\in\mathbb{R}$ represent the density, the horizonal velocity and vertical velocity respectively.

From the hydrostatic balance equation $(1.1)_{3}$, it follows that the density $\rho$ is independent of $z$. $\mu>0$, $\lambda\geq 0$ are the constant viscosity coefficients. The system is supplemented by the boundary conditions

[TABLE]

and initial data

[TABLE]

The pressure $p(\rho)$ satisfies the barotropic pressure law where the pressure and the density are related by the following formula:

[TABLE]

The PE model is widely used in meteorology and geophysical fluid dynamics, due to its accurate theoretical analysis and practical numerical computing. Concerning geophysical fluid dynamics we can refer to work by Chemin, Desjardins, Gallagher and Grenier [13] or Feireisl, Gallagher, Novotný [23]. There is a great number of results about PE, such as [3, 4, 9, 11, 12, 40, 41, 46, 48, 51]. We just mention some of results. Guillén-González, Masmoudi and Rodríguez-Bellido [28] proved the local existence of strong solutions. The celebrated breakthrough result was made by Cao and Titi [8]. They were first who proved the global well-posedness of PE. After that a lot of scientists were focused on the dynamics and regularity of PE e.g. [29, 30, 32, 33]. Recently in [9, 11, 12], the authors considered the strong solution for PE with vertical eddy diffusivity and only horizontal dissipation. About random perturbations of PE, the local and global strong solution of PE can be referred to [15, 16, 25], large deviation principles, see [17] and diffusion limit, see [31]. On the other hand, regarding to inviscid PE (hydrostatic incompressible Euler equations), the existence and uniqueness is an outstanding open problem. Only a few results are available. Under the convex horizontal velocity assumptions, Brenier [1] proved the existence of smooth solutions in two-dimensions. Then, Masmoudi and Wong [45] utilized the weighted $H^{s}$ a priori estimates and obtained the existence, uniqueness and weak-strong uniqueness. Removing the convex horizontal velocity assumptions, they extended Brenier’s result. By virtue of Cauchy-Kowalewshi theorem, the authors [36] constructed a locally, unique and real-analytic solution. Notably, Brenier [2] suggested that the existence problem may be ill-posed in Sobolev spaces. Further Cao et al. [10] established the blow up for certain class of smooth solutions in finite time.

In order to show the atmosphere and ocean have compressible property, Ersoy et al. [18] consider that the vertical scale of atmosphere is significantly smaller than the horizontal scales and they derive the CPE from the compressible Navier-Stokes equations. To be precise, the CPE system is obtained by replacing the vertical velocity momentum equation with hydrostatic balance equation. Compared with compressible Navier-Stokes equations, the regularity of vertical velocity is less regular than horizontal velocity in the CPE system. In the absence of sufficient information about the vertical velocity, it inevitably leads to difficulty for obtaining the existence of solutions. * Lions, Teman and Wang [38, 39] were first to study the CPE and received fundamental results in this field.* Under a smart $P-coordinates$, they reformulated the system into the classical PE with the incompressible condition. Later on, Gatapov and Kazhikhov [26], Ersoy and Ngom [19] proved the global existence of weak solutions in 2D case. Liu and Titi [42] used the classical methods to proved the local existence of strong solutions in 3D case. Ersoy et al. [18], Tang and Gao [47] showed the stability of weak solutions with the viscosity coefficients depending on the density. The stability means that a subsequence of weak solutions will converge to another weak solutions if it satisfies some uniform bounds. Recently, based on the work [4, 5, 6, 37, 49], Liu and Titi [43] and independently Wang et al. [50] used the B-D entropy to prove the global existence of weak solutions in the case where the viscosity coefficients are depending on the density.

Our paper is divided into two parts. The first part concerns the weak-strong uniqueness of CPE. Recently, Liu and Titi [44] studied the zero Mach number limit of CPE, proving it converges to incompressible PE, which is a breakthrough result to bridge the link between CPE and PE system. In the second part, inspired by [44], we investigate the singular limit of CPE, showing it converges to incompressible inviscid PE system. This is the first attempt to use the relative entropy method to study asymptotic limit for CPE. Let us mention that the corner-stone analysis of our results is based on the relative energy inequality which was invented by Dafermos, see [14]. It was introduced by Germain [27] and generalized by Feireisl [21] for compressible fluid model. Feireisl and his co-authors [22, 24] used the versatile tool to solve various problems. However, compared with the previous classical results, there is significant difference in the process of using relative energy inequality to CPE model due to the absence of the information on the vertical velocity. Therefore, it is not straightforward to apply the method from Navier-Stokes to CPE. We utilize the special structure of CPE to find the deeper relationship and reveal the important feature of CPE.

The paper is organized as follows. In Section 2, we introduce the dissipative weak solutions, relative energy and state our first theorem. In Section 3, we prove the weak-strong uniqueness. We recall the target system, state the singular limit theorem and derive the necessary uniform bounds in Section 4. Section 5 is devoted to proof of the convergence in the case of well-prepared initial data.

Part I: Weak-Strong uniqueness In this part, we focus on the weak-strong uniqueness of the CPE system.

2 Preliminaries and main result

First of all, we should point out that a proper notion of weak solution to CPE has not been well understood. Recently, Bresch and Jabin [7] consider different compactness method from Lions or Feireisl which can be applied to anisotropical stress tensor similarly. They obtain the global existence of weak solutions if $|\mu-\lambda|$ are not too large. Let us state one of the possible definitions here.

2.1 Dissipative weak solutions

Definition 2.1**. **

We say that $[\rho,\mathbf{u},w]$ is a dissipative weak solution to the system of (1.4), supplemented with initial data (1.3) and pressure follows (1.4) if $\rho=\rho(x,t)$ and

[TABLE]

$\bullet$ the continuity equation

[TABLE]

holds for all $\varphi\in C^{\infty}_{c}([0,T)\times\Omega)$;

$\bullet$ the momentum equation

[TABLE]

holds for all $\varphi\in C^{\infty}_{c}([0,T)\times\Omega)$,

$\bullet$ the energy inequality

[TABLE]

holds for a.a $\tau\in(0,T)$, a arbitrary constant $\overline{\rho}$, where $P(\rho)=\rho\int^{\rho}_{1}\frac{p(z)}{z^{2}}dz$.

Moreover, as there is no information about $w$, so we need the following equation:

[TABLE]

where

[TABLE]

We should emphasize that (2.5) is the key step to obtain the existence of weak solution in [43, 50], which is inspired by incompressible case.

2.2 Relative entropy inequality

Motivated by [21, 22], for any finite weak solution $(\rho,\mathbf{u},w)$ to the CPE system, we introduce the relative energy functional

[TABLE]

where $r>0$, $\mathbf{U}$ are smooth “test” functions, $r$, $\mathbf{U}$ compactly supported in $\Omega$.

Lemma 2.1**. **

Let $(\rho,\mathbf{u},w)$ be a dissipative weak solution introduced in Definition 2.1. Then $(\rho,\mathbf{u},w)$ satisfy the relative entropy inequality

[TABLE]

Proof: From the weak formulation and energy inequality (2.2)-(2.4) we deduce

[TABLE]

Summing (2.6)-(2.10) together, we obtain

[TABLE]

2.3 Main result

We say that $(r,\mathbf{U},W)$ is a strong solution to the CPE system $(1.1)-(1.4)$ in $(0,T)\times\Omega$, if

[TABLE]

with initial data $r^{\frac{1}{2}}_{0}\in H^{2}(\Omega)$, $r_{0}>0$ and $\mathbf{U}_{0}\in H^{3}(\Omega)$.

Now, we are ready to state our first result.

Theorem 2.1**. **

Let $\gamma>6$, $(\rho,\mathbf{u},w)$ be a dissipative weak solution to the CPE system $(1.1)-(1.4)$ in $(0,T)\times\Omega$. Let $(r,\mathbf{U},W)$ be a strong solution to the same problem and emanating from the same initial data. Then,

[TABLE]

Remark 2.1**. **

Liu and Titi [42] obtained the local existence of strong solutions to CPE. It is important to point out that our result holds under more regularity than the strong solutions obtained in [42].

Section 3 is devoted to the proof of the above theorem.

3 Weak-strong uniqueness

The proof of Theorem 2.1 depends on the relative energy inequality by considering the strong solution $[r,\mathbf{U},W]$ as test function in the relative energy inequality (2.6).

3.1 Step 1

We write

[TABLE]

As $[r,\mathbf{U},W]$ is a strong solution, it is easy to obtain that

[TABLE]

Moreover, the momentum equation reads as

[TABLE]

implying that

[TABLE]

So we rewrite

[TABLE]

Thus, we obtain that

[TABLE]

Before estimating, we should recall the following useful inequality from [21]:

[TABLE]

Moreover, from [21], we learn that

[TABLE]

The main difficulty is to estimate the complicated nonlinear term $\int_{\Omega}\rho(w-W)(\mathbf{U}-\mathbf{u})\cdot\partial_{z}\mathbf{U}dxdz$, we rewrite it as

[TABLE]

According to [21, 35], we divide the second term on the right side of (3.3) into three parts

[TABLE]

where in the last inequality, we have used the following celebrated inequality from Feireisl [20]:

Lemma 3.1**. **

Let $2\leq p\leq 6$, and $\rho\geq 0$ such that $0<\int_{\Omega}\rho dx\leq M$ and $\int_{\Omega}\rho^{\gamma}dx\leq E_{0}$ for some $(\gamma>1)$ then

[TABLE]

where $C$ depends on $M$ and $E_{0}$.

On the other hand, we take (2.5) into the first term on the right hand of (3.3) and get

[TABLE]

In the following, we will estimate the terms on the right hand side of (3.5). We choose the most complicated terms as examples to estimate, the remaining terms can be analyzed similarly. Firstly, we deal with $\int_{\Omega}\rho\widetilde{\mathbf{u}}\partial_{z}\nabla_{x}\mathbf{U}\cdot(\mathbf{U}-\mathbf{u})dxdz$ in the following,

[TABLE]

where $\widetilde{\mathbf{U}}=\int^{z}_{0}\mathbf{U}(x,s,t)ds$.

Similar to the above analysis, we divide the term $J_{2}$ into three parts

[TABLE]

On the other hand, by virtue of Cauchy inequality, we obtain

[TABLE]

Secondly, we will tackle with another complicated nonlinear term $\int_{\Omega}\rho\widetilde{\mathbf{u}}\partial_{z}\mathbf{U}\cdot(\nabla_{x}\mathbf{U}-\nabla_{x}\mathbf{u})dxdz$. It is easy to rewrite it as

[TABLE]

where

[TABLE]

Then we will deal with the second term on the right side of (3.7):

[TABLE]

where

[TABLE]

Next, by virtue of Hölder inequality, we get

[TABLE]

where we have used the interpolation inequality

[TABLE]

According (3.2) and (3.6), we have

[TABLE]

and

[TABLE]

Similar to the estimate of (3.6), we obtain

[TABLE]

Combining the above estimates, we get

[TABLE]

where $h(t)\in L^{1}(0,T)$.

Using the same method we estimate the remaining terms. Therefore, we conclude that

[TABLE]

Then we deduce that

[TABLE]

It is easy to check that

[TABLE]

where we have used the fact that $\partial_{t}r+\text{div}_{x}\mathbf{U}r+\mathbf{U}\cdot\nabla_{x}r+r\partial_{z}W=0$.

Recalling the boundary condition $W|_{z=0,1}=0$, we have

[TABLE]

Moreover, we can use the method as [35] Section 6.3 to get

[TABLE]

Putting $(3.10)-(3.13)$ together, we have

[TABLE]

Then applying the Gronwall’s inequality, we finish the proof of Theorem 2.1.

Part II: Singular limit of CPE

This part is devoted to studying the singular limit of the CPE in the case of well-prepared initial data.

4 Preliminaries and main result

From the notable survey paper by Klein, see [34], singular limits of fluids play an important role in mathematics, physics and meteorology. We consider the following scale CPE system with Coriolis forces:

[TABLE]

where $\epsilon$ represents the Mach number, $\omega=(0,0,1)$ is the rotation axis. The boundary conditions and pressure are the same as (1.2) and (1.4). Problem (4.4) is supplemented with initial data

[TABLE]

where the constant $\overline{\rho}$ in (4.2) can be taken arbitrary.

There is a quite broad consensus that the compressible flows become incompressible in the low Mach number limit. In the following sections, we assume $\rho=\rho_{\epsilon}$ and $\mathbf{u}=\mathbf{u}_{\epsilon}$. In this part, our goal is to study system (4.4) in the case of singular limit $\epsilon\rightarrow 0$, meaning the inviscid, incompressible limit. Precisely speaking, we want to show that the weak solutions of CPE converge to the incompressible PE system.

4.1 Target equation

The expected limit problem reads

[TABLE]

where $\mathbf{V}^{\perp}=(v_{2},-v_{1})$ and the $\Pi$ is the pressure. We supplement the system with the initial condition

[TABLE]

As shown by Kukavica et al. [33], the problem (4.6) possesses a local unique analytic solution $\mathbf{V}$ and $\Pi$ for some $T>0$ and any initial solution

[TABLE]

4.2 Relative energy inequality

According to the previous definition, we define the relative entropy functional,

[TABLE]

where $r$ and $\mathbf{V}$ are continuously differentiable, it is something not understandable ”text functions”. The following relation can be deduced

[TABLE]

for any r,\mathbf{V}$$\in C^{\prime}([0,T]\times\Omega), $r>0$.

4.3 Main result

The second result concerns the singular limit.

Theorem 4.1**. **

Let $\gamma>6$, and $(\rho,\mathbf{u},w)$ be a weak solution of the scaled system (4.4) on a time interval $(0,T)$ with well-prepared initial data satisfying the following assumptions

[TABLE]

Let $\mathbf{V}$ be the unique analytic solution of the target problem (4.6). Suppose that $T<T_{\max}$, where $T_{\max}$ denotes the maximal life-span of the regular solution to the incompressible PE system (4.6) with initial data $\mathbf{V}_{0}$, then

[TABLE]

where the constant $C$ depends on the initial data $\rho_{0}$, $\mathbf{u}_{0}$, $\mathbf{V}_{0}$ and $T$, and the size $D$ of the initial data perturbation. The constant $\overline{\rho}$ can be taken arbitrary.

Remark 4.1**. **

Theorem 4.1 yields that $\rho_{\epsilon}$ and $\mathbf{u}_{\epsilon}$ converge to the solution of target system in the regime of $\epsilon\rightarrow 0$ and $\mu,\lambda\rightarrow 0$ for the well-prepared initial data, in other words, the expression of the right hand of (4.8) tends to zero.

4.4 Uniform bounds

Before proving Theorems 4.1, we derive uniforms bounds of weak solutions $(\rho,\mathbf{u})$. Here and hereafter, the constant $C$ denotes a positive constant, independent on $\epsilon$, that will not have the same value when used in different parts of text. The following uniform bounds are derived from the relative energy inequality (4.9), if we take $r=\overline{\rho}$ and $\mathbf{U}=0$:

[TABLE]

5 Convergence of well-prepared initial data

The proof of convergence is based on the ansatz

[TABLE]

in the relative energy inequality (4.9), where $\mathbf{V}$ is the analytic solution of the target problem (4.6). The corresponding relative energy inequality reads as:

[TABLE]

First we deal with initial data and viscous term. It is easy to computer the initial relative energy inequality:

[TABLE]

and viscous term

[TABLE]

Next, we consider the remaining terms. Utilizing $(4.3)_{1}$, we get that

[TABLE]

It is easy to check that

[TABLE]

Next, we estimate the term $\int^{\tau}_{0}\int_{\Omega}\rho\mathbf{V}\cdot\nabla_{x}\Pi dxdzdt$, and rewrite in the form

[TABLE]

The second term on the right side of (5.6) is estimated as:

[TABLE]

where we have used the fact that $\Pi$ is independent of $z$. We deduce from the energy inequality that

[TABLE]

Similar to the previous analysis, it is enough to establish a uniform bound

[TABLE]

As we know that the pressure $\Pi$ is analytic, so that the rightmost integral of (5.6) can be vanished as $\epsilon\rightarrow 0$.

From the previous definition of dissipative weak solutions, we choose $\Pi$ as the test function, so that

[TABLE]

Compared with Navier-Stokes equations, the pressure term in PE system is easy to estimate. By virtue of incompressible condition and $(4.3)_{3}$, we have that

[TABLE]

Moreover, we find that

[TABLE]

Now, utilizing (2.5), we deal the complex nonlinear term

[TABLE]

These nonlinear terms are estimated one by one

[TABLE]

From the incompressible condition, it follows that $W=-\int^{z}_{0}\text{div}_{x}\mathbf{V}(x,s,t)ds$. We define $\widetilde{\mathbf{V}}=\int^{z}_{0}\mathbf{V}(x,s,t)ds$ and get

[TABLE]

The foremost two terms on the right side of (5.9) can be handed as (3.7)

[TABLE]

On the other hand, following (3.8), we have

[TABLE]

Similarly, the second nonlinear term on the right side of (5.8) is divided into two parts:

[TABLE]

Utilizing the similar estimates in (3.6), we have

[TABLE]

Moreover, similar to (3.4), we get

[TABLE]

and

[TABLE]

The last term can be estimated as

[TABLE]

Combining the above estimates together and using Grownwall inequality, we prove Theorem 4.1.

Acknowledgements

We are very much indebted to an anonymous referee for many helpful suggestions. The research of H. G is partially supported by the NSFC Grant No. 11531006. The research of Š.N. leading to these results has received funding from the Czech Sciences Foundation (GAČR), GA19-04243S and RVO 67985840. The research of T.T. is supported by the NSFC Grant No. 11801138. The paper was written when Tong Tang was visiting the Institute of Mathematics of the Czech Academy of Sciences which hospitality and support is gladly acknowledged.