Incompressible and vanishing vertical viscosity limit for the compressible Navier-Stokes system with Dirichlet boundary conditions

TL;DR

This paper proves the convergence of solutions from a compressible Navier-Stokes system with anisotropic dissipation to an incompressible system as vertical viscosity and Mach number vanish, handling complex boundary layers and oscillations.

Contribution

It establishes uniform regularity estimates and convergence results for the compressible Navier-Stokes system with anisotropic dissipation under Dirichlet boundary conditions, including ill-prepared initial data.

Findings

Solutions converge to incompressible system with horizontal dissipation

Uniform estimates independent of Mach number and vertical viscosity

Handling of boundary layers and oscillations in the limit process

Abstract

In this paper, we show the incompressible and vanishing vertical viscosity limits for the strong solutions to the isentropic compressible Navier-Stokes system with anistropic dissipation, in a domain with Dirichlet boundary conditions in the general setting of ill-prepared initial data. We establish the uniform regularity estimates with respect to the Mach number $\epsilon$ and the vertical viscosity $\nu$ so that the solution exists on a uniform time interval $[0,T_0]$ independent of these parameters. The key steps toward this goal are the careful construction of the approximate solution in the presence of both fast oscillations and two kinds of boundary layers together with the stability analysis of the remainder. In the process, it is also shown that the solutions of the compressible systems converge to those of the incompressible system with only horizontal dissipation, after removing the fast waves whose horizontal derivative is bounded in $L_{T_0}^2L_x^2$ by $\min\{1, (\epsilon/\nu)^{\frac14}\}.$