Low Mach number limit for the compressible Navier\u2013Stokes equations with density-dependent viscosity and vorticity-slip boundary condition

Dandan Ren,Xinfeng Liang,Yunting Ding

doi:10.1186/s13661-020-01455-9

Abstract

In this paper, we consider the three-dimensional compressible Navier–Stokes equations with density-dependent viscosity and vorticity-slip boundary condition in a bounded smooth domain. The main idea is to derive the uniform estimates for both time and the Mach number. The difficulty is dealing with density-dependent viscosity terms carefully. With the uniform estimates, we can verify the low Mach limit of the global strong solutions of compressible Navier–Stokes equations and the global existence and uniqueness of the strong solution of incompressible Navier–Stokes equations around a steady state.

Highlights

1 Introduction In this paper, we study the low Mach number limit for the initial-boundary value problem of the following three-dimensional compressible Navier–Stokes equations in a bounded domain ⊂ R3 with a smooth boundary: ρt + div(ρu) = 0, (1)
Concerning with the low mach number limit of the compressible non-isentropic Navier–Stokes equations, many results was presented in [2, 5, 12, 13, 15, 16, 20], and the references therein. After learning this progress on the low mach number limit carefully, we find the fact that most of it was concerned with the constant viscosity coefficients
We establish the uniform estimates of strong solutions with respect to the Mach number and justify rigorously the low Mach number limit for all time when the non-constant viscosity coefficients are present, in contrast with [21]

Summary

Introduction

We study the low Mach number limit for the initial-boundary value problem of the following three-dimensional compressible Navier–Stokes equations in a bounded domain ⊂ R3 with a smooth boundary: ρt + div(ρu) = 0,. In [21], Ou obtained the low Mach number limit of regular solutions to the compressible Navier–Stokes equations (1)– (2) with slightly compressible initial data in a 2-D bounded domain with the “vorticity-slip” boundary condition (6). The purpose of this paper is to verify rigorously the corresponding low Mach number limit for all time of the 3-D isentropic Navier–Stokes equations with density-dependent viscosity and the “vorticity-slip” boundary condition. We establish the uniform estimates of strong solutions with respect to the Mach number and justify rigorously the low Mach number limit for all time when the non-constant viscosity coefficients are present, in contrast with [21].

Cη u

Cη σt