On singular limit for the compressible Navier–Stokes system with non-monotone pressure law
Introduction
In this paper, we investigate the following scaled Navier–Stokes equations in the Eulerian coordinates: in . Here , and represent the density and the velocity respectively. The symbol denotes the barotropic pressure. The viscous stress is given by Newton’s rheological law where and are the viscosity coefficients. The Navier–Stokes system (1.1) is written in its dimensionless form, with the Mach number Ma and the Reynolds number . The system is supplemented by the far field conditions and initial conditions
In order to show the relative energy inequality, we introduce the function . We should emphasize that our results focus on the non-monotone pressure, which satisfies the following conditions: 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 and assumption (1.5), it is easy to check that On the other hand, from the condition , we can deduce that there exists small enough such that Therefore, similar to [1], [2], we can obtain
Our goal is to study the simultaneous singular limit and identify the target system as . 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 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 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 is a nonnegative function with the following regularity the continuity equation holds in the weak sense, specially for all ; the momentum equation is also satisfied in the sense of distributions,
Target equation
From Feireisl’s seminal work [1], [6], the target system is assumed to be the incompressible Euler system, which reads as: where is the limit velocity. We supplement the initial condition As shown by Kato’s work [23] and Oliver’s work [24], the problem (3.1)–(3.2) possesses a unique classical solution for some and any initial solution
Initial data
Before showing our theorems, we give two
Uniform bounds
Before proving theorems, we should derive some uniforms bounds for weak solutions . Here and hereafter, the constant denotes a positive constant, independent of , 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 as where
The uniform bounds are
Convergence of well-prepared initial data
The proof of convergence is based on the ansatz in the relative energy inequality (2.5), where is the smooth solution of the target problem (3.1).
The corresponding relative energy inequality:
First we deal with initial data and viscous term
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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The research of T.T. is supported by the NSFC Grant No. 11801138 and the Fundamental Research Funds for the Central Universities B200202156.