On singular limit for the compressible Navier–Stokes system with non-monotone pressure law

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Abstract

We consider a singular limit for the compressible Navier–Stokes system with general non-monotone pressure law in the asymptotic regime of low Mach number and large Reynolds numbers. We show that any dissipative weak solution approaches the solution of incompressible Euler equation both for well-prepared initial data and ill-prepared initial data.

Introduction

In this paper, we investigate the following scaled Navier–Stokes equations in the Eulerian coordinates: tρϵ+div(ρϵuϵ)=0,t(ρϵuϵ)+div(ρϵuϵuϵ)+1ϵ2xp(ρϵ)=νdivS(xuϵ),in (0,T)×Ω. Here Ω=R3, ρ=ρ(t,x) and u=u(t,x) represent the density and the velocity respectively. The symbol p(ρ) denotes the barotropic pressure. The viscous stress S is given by Newton’s rheological law S(xuϵ)=μ(xuϵ+xtuϵ)+λdivuϵI,μ>0,μ+λ0,where μ and λ are the viscosity coefficients. The Navier–Stokes system (1.1) is written in its dimensionless form, with the Mach number Ma =ϵ and the Reynolds number =ν1. The system is supplemented by the far field conditions uϵ0,ρϵρ¯,as|x|,whereρ¯>0,and initial conditions ρu|t=0=m0(x),ρ|t=0=ρ0(x).

In order to show the relative energy inequality, we introduce the function P(ρ)=ρρ¯ρp(z)z2dz. We should emphasize that our results focus on the non-monotone pressure, which satisfies the following conditions: pC1[0,)C2(0,),p(ρ)0,p(0)=0,lim infρP(ρ)p(ρ)>0,ρρ¯ρp(z)p(ρ¯)z2dz>0,whenρρ¯. We construct some examples to support our assumptions in the Appendix. It should be emphasized that our examples, in some sense, look like the van-der-Waals function, which has its own physical features. Under the definition of P(ρ) and assumption (1.5), it is easy to check that P(ρ)P(ρ¯)(ρρ¯)P(ρ¯)=ρρ¯ρp(z)p(ρ¯)z2dz>0,ρP(ρ)=p(ρ).On the other hand, from the condition ρρ¯ρp(z)p(ρ¯)z2dz>0, we can deduce that there exists δ>0 small enough such that p(ρ)>0when|ρρ¯|<δ.Therefore, similar to [1], [2], we can obtain P(ρ)P(ρ¯)(ρρ¯)P(ρ¯)C|ρρ¯|2,when|ρρ¯|<δ,C(1+p(ρ)),otherwise.

Our goal is to study the simultaneous singular limit and identify the target system as ϵ,ν0. The limit process is considered both for well-prepared initial data case and ill-prepared initial data case. The majority previous researches are concerned with the monotone pressure, such as standard polytropic pressure law, and a great number of weak–strong uniqueness results have been obtained. The key method is based on the relative energy inequality. It was Dafermos [3] who first introduced the concept of relative energy inequality then developed by Germain [4], Feireisl and his coauthors [1], [5], [6], [7], [8]. For using the relative energy inequality in other contexts, readers can refer to [9], [10], [11], [12], [13], [14], [15], [16]. The relative energy inequality offers us a powerful and elegant tool for the purpose of measuring the weak solution compared to another smooth solution. It is well-known that the monotone pressure is vital ingredient in the analysis of relative energy inequality. More precisely, the monotone pressure offers the convexity of the relative energy inequality. However, the research of two phase flow, such as liquid–vapor phase transformation, shows that the pressure is non-monotone. Specially speaking, the pressure function is assumed to be monotone increasing in some interval, then monotone decreasing in the other interval, and again monotone increasing in another interval (See [17] [Section 2]). Therefore, it is natural and interesting to investigate the relative energy inequality with non-monotone pressure law. Recently, Feireisl [18], Chaudhuri [19] and Giesselmann et al. [20] consider the weak–strong uniqueness problem for the compressible fluid with general non-monotone pressure law. We suggest readers can refer to [21], [22] for physical background and mathematical results about non-monotone pressure. This work can be seen as a extension and supplement of the previous work in the context of non-monotone pressure.

The paper is organized as follows. In Section 2, we introduce the definition of dissipative weak solutions, relative energy inequality and the other necessary material. We state our main theorems in Section 3 and derive uniform bounds of the Navier–Stokes system independent of ϵ in Section 4. Section 5 is devoted to proving the convergence of well-prepared initial data. In Section 6, we perform the necessary analysis of the acoustic waves and complete the proof of convergence of ill-prepared initial data.

Section snippets

Dissipative weak solutions

Definition 2.1

We say that [ρ,u] is a dissipative weak solution to the system of (1.1), supplemented with initial data (1.4) and pressure follows (1.5) if

the density ρ is a nonnegative function with the following regularity p(ρ)L(0,T;L1(Ω)),uL2(0,T;H1(Ω));ρ|u|2L(0,T;L1(Ω)),ρρ¯L(0,T;L2(Ω)),

the continuity equation holds in the weak sense, specially [Ωρφdx]t=0t=τ=0τΩρtφ+ρuxφdxdt,for all φCc([0,T)×Ω);

the momentum equation is also satisfied in the sense of distributions, [Ωρuφdx]t=0t=τ=0τΩρu

Target equation

From Feireisl’s seminal work [1], [6], the target system is assumed to be the incompressible Euler system, which reads as: divv=0,tv+vxv+xΠ=0, where v is the limit velocity. We supplement the initial condition v|t=0=v0.As shown by Kato’s work [23] and Oliver’s work [24], the problem (3.1)–(3.2) possesses a unique classical solution vC([0,Tmax);Wk,2(Ω)),tv,tΠ,xΠC([0,Tmax);Wk1,2(Ω)),k>3,for some Tmax>0 and any initial solution v0Wk,2(Ω),divv0=0.

Initial data

Before showing our theorems, we give two

Uniform bounds

Before proving theorems, we should derive some uniforms bounds for weak solutions (ρϵ,uϵ). Here and hereafter, the constant C denotes a positive constant, independent of ϵ, that will not have the same value when used in different parts of text. It is convenient to introduce the essential and residual part of an integrable function h as h(ρ,u)=[h]ess(ρ,u)+[h]res(ρ,u),[h]ess=ψ(ρ)h(ρ,u),[h]res=(1ψ(ρ))h(ρ,u),where ψCc(0,),0ψ(ρ)1,ψ(ρ)=1for allρ[12minR2ρ¯,2maxR2ρ¯].

The uniform bounds are

Convergence of well-prepared initial data

The proof of convergence is based on the ansatz r=ρ¯,U=v,in the relative energy inequality (2.5), where v is the smooth solution of the target problem (3.1).

The corresponding relative energy inequality: E(ρϵ,uϵ|ρ¯,v)|t=0t=τ+ν0τΩ(S(xuϵ)S(xv)):(xuϵxv)0τΩρ(tv+uϵxv)(vuϵ)dxdt+ν0τΩS(xv):(xvxuϵ).

First we deal with initial data E(ρϵ,uϵ|ρ¯,v)|t=0CΩ[|u0,ϵv0|2+|ρ0,ϵρ¯|2]dx,and viscous term νΩS(xv):(xvxuϵ)dx=2νΩS(xv):S(xvxuϵ)dxνS(xv)L22+νS(x(vuϵ))L22νS(xv)L22+ν

The ill-prepared initial data

This section is to perform the singular limit for the ill-prepared data as claimed in Theorem 3.2. It is well-known that ill-prepared data give rise to rapidly oscillating acoustic waves. Before our proof, we recall some classical work about acoustic equation.

Acknowledgments

The authors are grateful to the referees and the editors whose comments and suggestions greatly improved the presentation of this paper. The paper was written when Tong Tang was visiting the Institute of Mathematics of the Czech Academy of Sciences , and hospitality and support are gladly acknowledged. The author thanks Professor Feireisl for helpful discussion and constructive suggestions. Moreover, Doctor Burguet’s help is also gladly acknowledged.

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