Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary

Linjie Xiong

doi:10.3934/krm.2018021

Abstract

This paper concerns the low Mach number limit of weak solutions to the compressible Navier-Stokes equations for isentropic fluids in a bounded domain with a Navier-slip boundary condition. In [ 2 ], it has been proved that if the velocity is imposed the homogeneous Dirichlet boundary condition, as the Mach number goes to 0, the velocity of the compressible flow converges strongly in \begin{document}$ L^2$\end{document} under the geometrical assumption (H) on the domain. We justify the same strong convergence when the slip length in the Navier condition is the reciprocal of the square root of the Mach number.

Highlights

This paper is devoted to the incompressible limit for weak solutions to the compressible isentropic Navier-Stokes equations
The key idea of [8] is to employ the so-called “group” method developed by Schochet [10] and Grenier [4] to show that the fast acoustic waves does not affect the incompressible limit, justified the weak convergence
For the whole space case, Desjardins and Grenier [1] investigated the dispersive properties of the acoustic waves and used the Strichartz estimate to show that the acoustic waves vanish as the Mach number goes to zero, justified the strong convergence

Summary

Introduction

This paper is devoted to the incompressible limit for weak solutions to the compressible isentropic Navier-Stokes equations. If λ1 is an eigenvalue of L1 with multiplicity greater than or equal to 2, more precisely, if there exists l = k such that λl,0 = λk,0 = λ, and λl,1 = λk,1 = λ1, we need an extra orthogonality condition as following: Let H1 = H1(λ1) be defined by. Under the above assumption conditions, when the isentropic compressible NavierStokes equations (1.1) equipped with the Navier-Slip boundary conditions (1.3), ρε converges to 1 in C([0, T ]; Lγ(Ω)) and uεconverges to uweakly in L2((0, T ) × Ω)d for all T > 0 and strongly if Ω satisfies (H). We will only construct the acoustic boundary layers which yields the strong convergences if Ω satisfies (H)

Boundary Conditions depending upon the Mach Number

Construction of boundary layer

Induction go to the order

Strong Convergence of Quε