Singular limits for compressible inviscid rotating fluids

TL;DR

This paper investigates the behavior of rotating, compressible inviscid fluids under singular limits, showing convergence to an incompressible system as the Mach and Rossby numbers tend to zero.

Contribution

It introduces a general class of dissipative solutions for scaled compressible Euler systems and proves their convergence to an incompressible inviscid limit.

Findings

Dissipative solutions converge to strong solutions of the incompressible system.

The limit behavior is characterized as horizontal motion in an infinite slab.

The analysis applies to a broad class of solutions, including weak solutions.

Abstract

We study singular limit for scaled barotropic Euler system modelling a rotating, compressible and inviscid fluid, where Mach and Rossby numbers are proportional to a small parameter $\epsilon$. If the fluid is confined to an infinite slab, the limit behaviour is identified as a horizontal motion of an incompressible inviscid system that is analogous to the Euler system. We consider a very general class of solutions, named dissipative solution for the scaled compressible Euler systems and will show that it converges to a strong solution of that incompressible inviscid system.

Equations

$${\partial }_t\varrho +{\text{div}}_x\mathbf{m}$$

$${\partial }_t\mathbf{m}+{\text{div}}_x(\frac{\mathbf{m}\otimes \mathbf{m}}{\varrho })+\frac{1}{{\text{Ma}}^2}{\nabla }_{x}p(\varrho )+\frac{1}{\text{Ro}}\mathbf{b}\times \mathbf{m}$$

$$\text{Ma}\approx ϵ,\text{Ro}\approx ϵ\text{ for }ϵ>0.$$

$$\begin{array}{r}p(\varrho )=a{\varrho }^{\gamma },a>0,\gamma >1.\end{array}$$

$$\mathbf{u}\cdot \mathbf{n}=0,\mathbf{n}=(0,0,\pm 1).$$

$$\varrho \to \bar{\varrho },\mathbf{u}\to 0\text{ as }|x_h|\to \infty .$$

$${\varrho }_{ϵ}(0,\cdot )={\varrho }_{ϵ,0},{\mathbf{m}}_{ϵ}(0,\cdot )={\mathbf{m}}_{ϵ,0}.$$

$$\begin{array}{r}\chi (\varrho )\in C_c^{\infty }(0,\infty ),0\le \chi \le 1,\chi (\varrho )=1\text{mbox}if\frac{\bar{\varrho }}{2}\le \varrho \le 2\bar{\varrho },\bar{\varrho }>0.\end{array}$$

$$\begin{array}{r}[H{]}_{\text{ess}}=\chi (\varrho )H(\varrho ,\mathbf{m}),[H{]}_{\text{res}}=(1-\chi (\varrho ))H(\varrho ,\mathbf{m}).\end{array}$$

$$\begin{array}{rl}\varrho (t,x_h,-x_3)=\varrho (t,x_h,x_3),& {\mathbf{v}}_h(t,x_h,-x_3)={\mathbf{v}}_h(t,x_h,x_3),\\ & v_3(t,x_h,-x_3)=-v_3(t,x_h,x_3).\end{array}$$

$$P(\varrho )=\varrho {\int }_{\bar{\varrho }}^{\varrho }\frac{p(z)}{z^2}\text{d}z.$$

$$\varrho P^{\prime }(\varrho )-P(\varrho )=p(\varrho )\text{ and }\varrho P^{\prime \prime }(\varrho )=p^{\prime }(\varrho )\text{ for }\varrho >0.$$

$${\varrho }_{ϵ}-\bar{\varrho }\in C_{\text{weak}}([0,T];L^2+L^{\gamma }(\Omega )),{\varrho }_{ϵ}\ge 0,{\mathbf{m}}_{ϵ}\in C_{\text{weak}}([0,T];L^2+L^{\frac{2\gamma }{\gamma +1}}(\Omega ))$$

$$\displaystyle\begin{split}\varrho_{0,{\epsilon}}\geq 0,\;E_{0,{\epsilon}}=\int_{\Omega}\bigg{(}\frac{1}{2}\frac{|\mathbf{m}_{0,{\epsilon}}|^{2}}{\varrho_{0,{\epsilon}}}+P(\varrho_{0,{\epsilon}})-(\varrho_{0,{\epsilon}}-\bar{\varrho})P^{\prime}(\bar{\varrho})\bigg{)}\text{ d}x<\infty,\end{split}$$

$$\begin{array}{rl}& {\mathfrak{R}}_{v_{ϵ}}\in L^{\infty }(0,T;{\mathcal{M}}^{+}(\Omega ;{\mathbb{R}}_{\text{sym}}^{d\times d})),{\mathfrak{R}}_{p_{ϵ}}\in L^{\infty }(0,T;{\mathcal{M}}^{+}(\Omega )),\end{array}$$

$$\displaystyle\begin{split}&\big{[}\int_{\Omega}{(\varrho_{\epsilon}-\bar{\varrho})}\varphi\text{ d}x\big{]}_{t=0}^{t=\tau}=\int_{0}^{\tau}\int_{\Omega}[\varrho_{\epsilon}\partial_{t}\varphi+\mathbf{m}_{\epsilon}\cdot\nabla_{x}\varphi]\text{ d}x\text{ d}t\;.\end{split}$$

$$\displaystyle\begin{split}&\bigg{[}\int_{\Omega}\mathbf{m}_{\epsilon}(\tau,\cdot)\cdot\boldsymbol{\varphi}(\tau,\cdot)\text{ d}x\bigg{]}_{t=0}^{t=\tau}\\ &=\int_{0}^{\tau}\int_{\Omega}[\mathbf{m}_{\epsilon}\cdot\partial_{t}\boldsymbol{\varphi}+(\frac{\mathbf{m}_{\epsilon}\otimes\mathbf{m}_{\epsilon}}{\varrho_{\epsilon}}):\nabla_{x}\boldsymbol{\varphi}+p(\varrho_{\epsilon})\text{div}_{x}\boldsymbol{\varphi}+\mathbf{b}\times\mathbf{m}_{\epsilon}\cdot\boldsymbol{\varphi}]\text{ d}x\text{ d}t\;\\ &+\int_{0}^{\tau}\int_{\Omega}\nabla_{x}\boldsymbol{\varphi}:\text{d}[\mathfrak{R}_{p_{\epsilon}}+\mathfrak{R}_{p_{\epsilon}}\mathbb{I}]\text{ d}t\;.\end{split}$$

$$E_{ϵ}(\tau )={\int }_{\Omega }$$

$$\displaystyle\begin{split}\bigg{[}\psi\bigg{(}E_{\epsilon}(\tau)+\int_{\Omega}\bigg{[}\frac{1}{2}\text{d}\text{ trace}\mathfrak{R}_{v_{\epsilon}}(\tau,\cdot)+\frac{1}{\gamma-1}\text{d}\mathfrak{R}_{p_{\epsilon}}(\tau,\cdot)\bigg{]}\bigg{)}\bigg{]}_{\tau=\tau_{1}-}^{\tau=\tau_{2}+}\\ \leq\int_{\tau_{1}}^{\tau_{2}}\partial_{t}(\psi)\bigg{(}E_{\epsilon}(t)+\int_{\bar{\Omega}}\frac{1}{2}\text{d}\text{ trace}\mathfrak{R}_{{v_{\epsilon}}}+\int_{\bar{\Omega}}\frac{1}{\gamma-1}\text{d}\mathfrak{R}_{{p_{\epsilon}}}\bigg{)}\text{ d}t\;\end{split}$$

$$\begin{array}{rl}& \frac{p^{\prime }(\bar{\varrho })}{\bar{\varrho }}{\nabla }_{x}q+\mathbf{b}\times \mathbf{v}=0,\\ & {\partial }_t({\Delta }_{x_h}q-\frac{1}{p^{\prime }(\bar{\varrho })}q)+({\nabla }_{x_h}^{\perp }q\cdot {\nabla }_{x_h})({\Delta }_{x_h}q-\frac{1}{p^{\prime }(\bar{\varrho })}q)=0,\text{ in }{\mathbb{R}}^2.\end{array}$$

$${\varrho }_{ϵ}=\bar{\varrho }+ϵ{\varrho }_{ϵ}^{(1)}+{ϵ}^2{\varrho }_{ϵ}^{(2)}+\cdots ,$$

$${\mathbf{m}}_{ϵ}=\bar{\varrho }\mathbf{v}+ϵ{\mathbf{m}}_{ϵ}^{(1)}+{ϵ}^2{\mathbf{m}}_{ϵ}^{(2)}+\cdots .$$

$$p({\varrho }_{ϵ})=p(\bar{\varrho })+ϵp^{\prime }(\bar{\varrho }){\varrho }_{ϵ}^{(1)}+{ϵ}^2(p^{\prime }(\bar{\varrho }){\varrho }_{ϵ}^{(2)}+\frac{1}{2}{p}^{\prime \prime }(\bar{\varrho })({\varrho }_{ϵ}^{(1)}{)}^2)+o({ϵ}^3).$$

$$\bar{\varrho }{\text{div}}_x\mathbf{v}+ϵ({\partial }_t{\varrho }_{ϵ}^{(1)}+{\text{div}}_x{\mathbf{m}}_{ϵ}^{(1)})+o({ϵ}^2)=0$$

$$\bar{\varrho }({\partial }_t\mathbf{v}+(\mathbf{v}\cdot {\nabla }_x)\mathbf{v})+{\nabla }_x(p^{\prime }(\bar{\varrho }){\varrho }_{ϵ}^{(2)}+\frac{1}{2}{p}^{\prime \prime }(\bar{\varrho })({\varrho }_{ϵ}^{(1)}{)}^2)+\mathbf{b}\times {\mathbf{m}}_{ϵ}^{(1)}$$

$$+\frac{1}{ϵ}(p^{\prime }(\bar{\varrho }){\nabla }_x{\varrho }_{ϵ}^{(1)}+\bar{\varrho }\mathbf{b}\times \mathbf{v})+o(ϵ)=0.$$

$$p^{\prime }(\bar{\varrho }){\nabla }_{x}q+\bar{\varrho }\mathbf{b}\times \mathbf{v}=0,$$

$${\text{div}}_x\mathbf{v}=0,$$

$$\bar{\varrho }({\partial }_t\mathbf{v}+(\mathbf{v}\cdot {\nabla }_x)\mathbf{v})+{\nabla }_x{\Pi }_1+\mathbf{b}\times {\mathbf{m}}_1=0,$$

$${\partial }_{t}q+{\text{div}}_x{\mathbf{m}}_1=0.$$

$$q_{x_3}=0,q(x)=q(x_h),{\nabla }_{x_h}^{\perp }q=\frac{\bar{\varrho }}{p^{\prime }(\bar{\varrho })}{\mathbf{v}}_h,{\mathbf{v}}_h=(v_1,v_2),$$

$${\text{div}}_{x_h}{\mathbf{v}}_h=0,v_{3_{x_3}}=0,\text{ here }{\nabla }_{x_h}^{\perp }\equiv (-\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_1}).$$

$$v_{1_{x_3}}=0,v_{2_{x_3}}=0$$

Full text

Technische Universität, Berlin

Institute für Mathematik, Straße des 17. Juni 136, D – 10623 Berlin, Germany.

Keywords: Compressible Euler system, rotating fluids, dissipative solution, low Mach and Rossby number limt.

AMS classification: Primary: 76U05; Secondary: 35Q35, 35D05, 76N10

1 Introduction

Several types of singular limits of Navier–Stokes system and Euler system have been studied in the last few decades. Here we devote ourselves on the models arising by the effect of rotation on fluids as described in Chemin et.al. [CDGG2006].

Let $T>0$ and $\Omega\subset(\mathbb{R}^{3})=\mathbb{R}^{2}\times(0,1)$ be an infinite slab. We consider the scaled compressible Euler equation in time-space cylinder $Q_{T}=(0,T)\times\Omega$ describing the time evolution of the mass density $\varrho=\varrho(t,x)$ and the momentum field $\mathbf{m}=\mathbf{m}(t,x)$ of a compressible rotating inviscid fluid with axis of rotation $\mathbf{b}=(0,0,1)$:

• •

**Conservation of Mass: **

[TABLE]

• •

Conservation of Momentum:

[TABLE]

• •

The scaled system contains charecteristic numbers:

Ma– Mach number,

Ro– Rossby number.

Here we consider,

[TABLE]

• •

Pressure Law: In an isentropic setting, the pressure $p$ and the density $\varrho$ of the fluid are interrelated by

[TABLE]

• •

Boundary condition: Here we consider slip condition for velocity on the horizontal boundary i.e.

[TABLE]

• •

Far field condition: Introducing the notation $x=(x_{h},x_{3})$ and $P_{h}(x)=x_{h}$ we assume this condition as,

[TABLE]

• •

Initial data: For each $\epsilon>0$, we supplement the initial data as

[TABLE]

Our goal here is to study the effect of low Mach number limit (also called incompressible limit) and low Rossby number limit on the system (1.1)-(1.2). In low Mach number limit fluid becomes incompressible and in low Rossby number limit indicates fast rotation of fluids as a consequence of that fluid becomes planner (two-dimensional). In our case when $\epsilon\rightarrow 0$, we have both phenomenon described above simultaneously, i.e the system (1.1)-(1.2) (primitive system) which describes a compressible, rotating (3D) fluid is expected to become a system (target system) that describes incompressible, planner(2D) fluid.

There are mainly two approaches to deal with the singular limit problem.

• I.

The first approach consists of considering a classical(strong) solution of the primitive system and expected that it converges to the classical solutions of the target system. Here, the main and highly nontrivial issue is to ensure that the lifespan of the strong solutions is bounded below away from zero uniformly with respect to the singular parameter.

• II.

The second approach is based on the concept of weak or measure–valued dissipative solutions of the primitive system. Under proper choice of initial data one can show convergence provided the target system admits smooth solution.

For the first approach in the low Mach number limits we have results by Schochet[S1986], Ebin [E1977], Kleinermann-Majda[KM1981] and many others. For rotating fluids there are results by Babin et. al [BMN1999], [BMN2001] and Chemin et. al. [CDGG2006]. Since it is well known fact that strong solution of compressible inviscid fluids develop singularities, the main difficulty here is to show the existence time of primitive system is independent of $\epsilon$.

In the case of second approach, a similar problem has been studied by Feireisl et. al. in [FN2014]. They consider viscous fluid(Navier-Stokes System) and also high Reynolds number limit along with low Mach and low Rossby number limit. They have proved that weak solution of their system converges to the strong solution of the same target system as we have . Relative energy inequality plays very important role in the proof. Also Feireisl et. al. [FGN2012] have studied the low Mach and Rossby number limit for scaled Navier-Stokes system. In this case they obtain they obtain a similar system with some effect of viscous term.

The advantage(s) to consider the second approach is(are),

• •

Weak or measure valued solutions to the primitive system exist globally in time. Hence the result depends only on the life span of the target problem that may be finite.

• •

The result convergence holds for a large class of generalized solutions which indicates certain stability of the limit solution of the target system.

Recently, the concept of measure–valued solutions has been studied in variuos context, like, analysis of numerical schemes etc. In the following articles [AB1997], [PAW2015], [BF2018b], [B2018], [Bd2019], [FL2018] we observe the development of theory on measure valued solution for different models describing compressible fluids mainly with the help of Young measures. But Feireisl et. al. in [FLM2019] and [BeFH2019] give a new definition termed as dissipative solution without involving Young measures. We will follow the last approach.

In Feireisl et. al. [FKM2019], Bruell et. al. [BrF2018] and Brézina et. al.[BM2018] show that measure valued solution of primitive system which describes some compressible inviscid fluid converges to strong solution of incompressible target system.

In second approach, it is very important to consider proper initial data mainly termed as well-prepared and ill-prepared initial data. Feireisl and Novotný in [FN2009b], explain that for ill-prepared data the presence of Rossby-acoustic waves play an important role in analysis of singular limits. Meanwhile this effect was absent in well-prepared data. Here we deal with both types of initial data.

Our main goal is to prove that a dissipative solution behaves similar to the weak solutions, i.e. in low Mach and low Rossby regime they converges to strong solution which descibes planner, incompressible fluid in 2D. Hence our plan for the article is,

Definition of dissipative solution.

• 2.

Singular limit for ‘well-prepared’ data.

• 3.

Singular limit for ‘ill-prepared’ data.

1.1 Notation:

• •

To begin, we introduce a function $\chi=\chi(\varrho)$ such that

[TABLE]

For a function, $H=H(\varrho,\mathbf{m})$ we set

[TABLE]

• •

Following the article by Feireisl and Novotný [FN2014] we can conclude that the consideration of the domain $\mathbb{R}^{2}\times(0,1)$ is equivalent to consider $\mathbb{R}^{2}\times\mathbb{T}^{1}$. Since, the complete slip boundary conditions can be transformed to periodic ones by considering the space of symmetric functions, the horizontal components of the velocity are even and the vertical one odd with respect to the vertical variable, i.e.

[TABLE]

for all $t\in(0,T),\;x_{h}\in\mathbb{R}^{2},\;x_{3}\in\mathbb{T}^{1}$. This equivalence has been described in Ebin[E1983]. So from now on we consider $\Omega=\mathbb{R}^{2}\times\mathbb{T}^{1}$.

• •

Let us define pressure potential as,

[TABLE]

As a consequence of that we have,

[TABLE]

2 Definition of dissipative solution

We first want to give a definition of dissipative solution for our choosen system. Basarić in [Bd2019] introduces a dissipative measure valued solution for Euler equation with damping by introducing Young measures in an unbounded domain. For bounded domain, Feireisl et. al.[FLM2019] and [BeFH2019] recently introduced a concept of dissipative solution without defining solution through Young Measures. We follow the second approach to define dissipative solution.

Definition 2.1**.**

Let $\epsilon>0$ and $\bar{\varrho}>0$. We say functions $\varrho_{\epsilon},\mathbf{m}_{\epsilon}$ with,

[TABLE]

and also satisfying (1.10) are a dissipative solution to the compressible Euler equation (1.1)-(1.6) with initial data $\varrho_{0,\epsilon},\mathbf{m}_{0,\epsilon}$ satisfying,

[TABLE]

if there exist the turbulent defect measures

[TABLE]

such that the following holds,

• •

Equation of Continuity: For any $\tau\in(0,T)$ and any $\varphi\in C_{c}^{1}([0,T]\times\bar{\Omega})$ it holds

[TABLE]

• •

Momentum equation: For any $\tau\in(0,T)$ and any $\boldsymbol{\varphi}\in C^{1}_{c}([0,T]\times\Omega;\mathbb{R}^{d})$, it holds

[TABLE]

• •

Energy inequality: The total energy $E$ is defined in $[0,T)$ as,

[TABLE]

It satisfies,

[TABLE]

for any $0\leq\tau_{1}\leq\tau_{2}<T$ and any $\psi\in C_{c}^{1}[0,T)$, $\psi\geq 0$.

Remark 2.2**.**

In (2.6), the initial energy satisfies $E_{\epsilon}(0-)=E_{0,\epsilon}$.

Theorem 2.3**.**

Suppose $\Omega$ be the domain specified above and pressure follows (1.4). If $(\varrho_{0,\epsilon},\mathbf{m}_{0,\epsilon})$ satisfies (2.2), then there exists dissipative solution as defined in definition (2.1).

The proof this theorem follows in similar lines of Breit, Feireisl and Hofmanová as in [BeFH2019] and [BeFH2019(2)]. We have to adopt it for unbounded domain as suggested in Basarić[Bd2019].

3 Identification of Target system

Taking motivation from [FN2014] we expect the target system as,

[TABLE]

3.1 Informal justification:

Here is an informal justification how we obtain the target system as in (3.1). First we note that $(\bar{\varrho},0)$ is a steady state solution for (1.1)-(1.6).

Let us consider

[TABLE]

As a consequence of the above we obtain,

[TABLE]

So we obtain,

[TABLE]

and

[TABLE]

Further we assume \big{(}\frac{\varrho_{\epsilon}-\bar{\varrho}}{\epsilon}\big{)}\rightarrow q and $\mathbf{m}_{\epsilon}\rightarrow\bar{\varrho}\mathbf{v}$ in some strong sense. Then as a consequence we have

[TABLE]

Few consequence from above relations,

[TABLE]

Assuming smoothness,

[TABLE]

So,

[TABLE]

Also boundary condition $\frac{\mathbf{m}_{\epsilon}}{\varrho_{\epsilon}}\cdot\mathbf{n}=0$ will lead to conclude

[TABLE]

Then we will have $\mathbf{v}=(\mathbf{v}_{h}(x_{h}),0)$. Finally we obtain,

[TABLE]

Here $q$ can be viewed as a kind of stream function. As a simple consequence of above we have, $\text{div}_{x_{h}}\mathbf{v}=0$.

3.2 Reguarity of Target System (3.1)

Since the (3.1) is possesses the same structure as 2D Euler equations. We expect solution to be as regular as the initial data and exists globally in time. In particular, we may use the abstract theory of Oliver[O1997], Theorem 3, to obtain the result:

Proposition 3.1**.**

Suppose that

[TABLE]

Then, the problem (3.1) admits a solution $q$, unique in the class

[TABLE]

4 Singular limit for "Well-prepared" initial Data

After giving a proper definition of well-prepared data we will state our mail result and will prove it in the following sections.

4.1 Definition of "Well-prepared Data"

Definition 4.1**.**

We say that the set of initial data $(\varrho_{0,\epsilon},\mathbf{m}_{0,\epsilon})_{(\epsilon>0)}$ is "well-prepared" if,

[TABLE]

For $\epsilon>0$, $(\varrho_{\epsilon},\mathbf{m}_{\epsilon})$ are solutions of (1.1)-(1.6) with initial data (1.7). If we have "Well–prepared initial data" then claim is $(\varrho_{\epsilon},\mathbf{m}_{\epsilon})\rightarrow(\bar{\varrho},\mathbf{v}_{h},q)$ where $(\bar{\varrho},\mathbf{v}_{h},q)$ is a classical solution of the (3.1) with initial data $(P_{h}(\mathbf{v}_{0}),0)$ in $\mathbb{R}^{2}$.

4.2 Main Theorem:

Theorem 4.2**.**

Let pressure $p$ follows (1.4). Further we assume that the initial data is well-prepared, i.e. it follows (4.1). Then after taking a subsequence, the following holds,

[TABLE]

Now if we assume that $\mathbf{v}_{0}\in W^{k,2}$ with $k\geq 3$. Then $(q,\mathbf{v})$ solves (3.1) with initial data $q(0,\cdot)=q_{0}$ where $q_{0}$ satisfies (4.1).

In the following subsections we will complete the proof.

4.3 Relative energy inequality

Relative energy plays an important role to conclude the proof of the theorem (4.2). For $\epsilon=1$, the relative energy defines as,

[TABLE]

where $\tilde{\varrho},\tilde{\mathbf{u}}$ are arbitrary smooth test functions with $\tilde{\varrho}-\bar{\varrho}$ and $\tilde{\mathbf{u}}$ have compact support .

Remark 4.3**.**

The relative energy is a coercive functional (see. Bruell et. al. [BrF2018]) satisfying the estimate,

[TABLE]

From (4.3) we can deduce that,

[TABLE]

Now using the dissipative solution we obtain,

[TABLE]

Finally we obtain,

[TABLE]

Now we consider the scaling on (4.3) and rephrase it as,

[TABLE]

We write the scaled version of the inequality as,

[TABLE]

We rewrite the above inequality as,

[TABLE]

Suppose, $(\tilde{\varrho}-\bar{\varrho},\tilde{\mathbf{u}})\in C^{1}([0,T];H^{m}(\Omega))\times C^{1}([0,T];H^{m}(\Omega))$ with $m$ large. We know $C^{1}([0,T];C_{c}^{\infty}(\Omega))$ is dense in Sobolev space $C^{1}([0,T];H^{m}(\Omega))$. For $\delta>0$ we have, $\tilde{r}\in C^{1}([0,T];C_{c}^{\infty}(\Omega))$ and $\tilde{\mathbf{v}}\in C^{1}([0,T];C_{c}^{\infty}(\Omega))$.

[TABLE]

Following Theorem 2.3 of [Bd2019] we can show that,

[TABLE]

Thus for Sobolev functions, relative energy inequality (4.7) is true.

4.4 Convergence: Part 1

First with $\tilde{\mathbf{u}}=0,\;\tilde{\varrho}=\bar{\varrho}$ as test functions we obtain the following bounds,

[TABLE]

As an immediate consequence of above we have,

[TABLE]

We obtain,

[TABLE]

and

[TABLE]

Thus we have,

[TABLE]

Now from these two we write, $\mathbf{m}=\bar{\varrho}\mathbf{u}$ and letting $\epsilon\rightarrow 0$ in the continuity equation we have,

[TABLE]

Furthermore multiplying momentum equation by $\epsilon$ and letting $\epsilon\rightarrow 0$ we get the diagonasticequation,

[TABLE]

in the sense of distributions.

Clearly from last relation we have $\varrho^{(1)}$ is independent of $x_{3}$, i.e. $\varrho^{(1)}=\varrho^{(1)}(x_{h})$. Thus,

[TABLE]

Now from the symmetry class in which $\mathbf{u}$ belongs we have,

[TABLE]

4.5 Convergence: Part 2

We recall the target system here,

[TABLE]

Let $(q,\mathbf{v})$ be solution of the above system with initial data $q_{0},\mathbf{v}_{0}$ satisfying (4.1). Our goal is to show that $(\varrho^{(1)},\mathbf{u})\equiv(q,\mathbf{v})$. Here we choose proper test functions and will show that $\lim\limits_{\epsilon\rightarrow 0}\mathcal{E}_{\epsilon}(t)=0$.

We consider,

[TABLE]

We rewrite the relative energy inquality as,

[TABLE]

Using the fact $\text{div}_{x_{h}}\mathbf{v}=0$, we obtain,

[TABLE]

Consideration of well prepared data yields,

[TABLE]

Hence we conclude,

[TABLE]

Here $c(\epsilon)$ is a generic function such that $c(\epsilon)\rightarrow 0$ as $\epsilon\rightarrow 0$.

We write,

[TABLE]

and

[TABLE]

Now using above and results previous part, we obtain,

[TABLE]

Now using properties of $q,\mathbf{u}$ and $\mathbf{v}$ we can conclude that,

[TABLE]

since,

[TABLE]

Clearly the third term vanishes. Now it is enough to prove,

[TABLE]

The above inequality is true by multiplying second equation of target system by $q$.

Hence we have,

[TABLE]

Now we also obtain,

[TABLE]

Next we consider,

[TABLE]

It is easy to verify that,

[TABLE]

From here we can conclude that,

[TABLE]

Hence we have

[TABLE]

Also we obtain,

[TABLE]

Combining all these (4.14)-(4.18) we have,

[TABLE]

Using Gronwall lemma we have,

[TABLE]

where $c(\epsilon)\rightarrow 0$ as $\epsilon\rightarrow 0$. Hence,

[TABLE]

Now using coerceivity of relative energy functional we can say,

[TABLE]

From (4.21) we can conclude

[TABLE]

It ends the proof of the theorem.

5 Singular limit for "ill-prepared data"

5.1 Definition of ill-prepared data

We define the ill-prepared initial data as,

Definition 5.1**.**

We say that the set of initial data $(\varrho_{0,\epsilon},\mathbf{m}_{0,\epsilon})_{(\epsilon>0)}$ is "well-prepared" if,

[TABLE]

5.2 Main Theorem:

Theorem 5.2**.**

Let pressure $p$ follows (1.4). Further we assume that the initial data is ill-prepared, i.e. it follows (5.1). Also we assume intial data has better regularity,

[TABLE]

Then after taking a subsequence, the following holds,

[TABLE]

Then $(q,\mathbf{v})$ solves (3.1) with initial data $q(0,\cdot)=q_{0}$ where $q_{0}$ is the solution of elliptic equation,

[TABLE]

One major problem for the analysis of the singular limit is the presence of rapidly oscillating Rossby-acoustic waves. Although these acoustic waves disappear in the course of low-mach number limit.

5.2.1 Informal justification of derivation of the acoustic wave equation:

We rewrite the equation in fast time scale $\tau=\frac{t}{\epsilon}$ as,

[TABLE]

[TABLE]

Again considering,

[TABLE]

and comparing zeroth order term for continuity equation and $o(\epsilon^{-1})$ for momentum equation, we have,

[TABLE]

Going back to previous time scale we have,

[TABLE]

This motivates us how we obtain this hyperbolic system which represents acoustic waves.

5.3 Proof of the theorem:

The proof is in similar lines as in well-prepared case The first part is quite similar as we have done for the well-prepared data in Convergence: part 1. Now for second part we need some extra term and dissipative estimate to deal it. We recall the same relative energy ineqality,

[TABLE]

We rewrite the above one as,

[TABLE]

5.3.1 Initial data decomposition

Following [FN2014] subsection 4.3 we will regularize initial data $\varrho_{0}^{(1)},\;\mathbf{u}_{0}=\frac{\mathbf{m}_{0}}{\varrho_{0}}$ as,

[TABLE]

with the help of cut-off function and Fourier transforms to gain both integrability and smoothness. Now decompose

[TABLE]

with

[TABLE]

Here we consider $s_{\epsilon,\delta},\;\mathbf{V}_{\epsilon,\delta}$ solve

[TABLE]

with initial data $s_{0,\delta},\;\mathbf{V}_{0,\delta}$.

Also cosider $q_{\delta},\mathbf{v}_{\delta}$ solve

[TABLE]

with initial data $q_{0,\delta},\mathbf{v}_{0,\delta}$.

Next we recall the result from [FN2014] Section 4 which is one of the important result concerning dispersive estimates of acoustis-rossby waves.

Proposition 5.3**.**

For the choice of initial data $s_{0,\delta},\;\mathbf{V}_{0,\delta}$ as above we have the following result when $\epsilon\rightarrow 0$,

[TABLE]

for any fixed $\delta>0$, any $1\leq p<\infty$ and $m=0,1,2,\cdots$.

5.3.2 Relative Energy inequality and Convergence: Part 2

First we have same apriori bounds and convergence of $\varrho_{\epsilon},\mathbf{m}_{\epsilon}$ as in well-prepared case (See Convergence: Part 1). Now we need to choose a different test functions in this case.

Considering

[TABLE]

we rewrite the previous inequality as,

[TABLE]

Now,

[TABLE]

Further we have,

[TABLE]

Hence we have,

[TABLE]

We introduce a notaion $h_{i}(\epsilon,\delta)$ as $\lim\limits_{\epsilon\rightarrow 0}h_{i}(\epsilon,\delta)=\bar{h}_{i}(\delta)$ and $\lim\limits_{\delta\rightarrow 0}\bar{h}_{i}(\delta)=0$ for $i\in\mathbb{N}$.

[TABLE]

[TABLE]

[TABLE]

Eventually we have,

[TABLE]

[TABLE]

For $\mathcal{T}_{5}$, we use similar treatment as $\mathcal{T}_{2}$ we have

[TABLE]

with $|\bar{\mathcal{T}}_{5}|\leq h_{5}(\epsilon,\delta),\text{ with }\bar{h}_{5}(\delta)=0$ and

[TABLE]

Following similar reasoning line as in well-prepared case, we have,

[TABLE]

for $\mathcal{T}_{6}$ we have,

[TABLE]

with $|\bar{\mathcal{T}}_{6}|\leq h_{6}(\epsilon,\delta),\text{ with }\bar{h}_{6}(\delta)=0,$ and

[TABLE]

We also have,

[TABLE]

Thus we have

[TABLE]

Passing limit for first for $\epsilon\rightarrow 0$, then for $\delta\rightarrow 0$ we can conclude the proof of theorem.

6 Concluding Remarks:

Clearly we can see well-prepared case as a particuler case of ill prepared case. Although here we show that for well prepared case you dont need any dispersive estimates.

Acknowledgement

The work is supported by Einstein Stiftung, Berlin. I would like to thank my Ph.D supervisor Prof. E. Feireisl for his valuable suggestions and comments.