On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain

TL;DR

This paper extends Kato's inviscid limit results from incompressible to compressible Navier-Stokes equations in bounded domains, analyzing boundary conditions and boundary layer effects.

Contribution

It generalizes Kato's boundary layer convergence criteria to the compressible case and examines slip boundary conditions with friction.

Findings

Inviscid limit holds under vanishing energy dissipation in boundary layers.

Convergence to Euler equations is valid with slip boundary conditions and moderate friction.

Uses recent relative energy estimates to establish results.

Abstract

In this paper we investigate the issue of the inviscid limit for the compressible Navier-Stokes system in an impermeable fixed bounded domain. We consider two kinds of boundary conditions. The first one is the no-slip condition. In this case we extend the famous conditional result obtained by Kato in the homogeneous incompressible case. Kato proved that if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes then the solutions of the incompressible Navier-Stokes equations converge to some solutions of the incompressible Euler equations in the energy space. We provide here a natural extension of this result to the compressible case. The other case is the Navier condition which encodes that the fluid slips with some friction on the boundary. In this case we show that the convergence to the Euler equations holds true in the energy space, as least when the friction is not too large. In both cases we use in a crucial way some relative energy estimates proved recently by Feireisl, Ja Jin and Novotn{\'y}.