Remarks on the inviscid limit for the compressible flows

TL;DR

This paper investigates the conditions under which compressible Navier-Stokes flows converge to Euler flows in the inviscid limit, extending known criteria to general domains with boundaries and including classical boundary conditions.

Contribution

It extends Kato's criteria for the inviscid limit to compressible flows in general domains with boundary conditions, including no-slip and Navier types, and establishes convergence results.

Findings

Extended Kato criteria to compressible flows

Proved convergence in the relative energy norm for smooth solutions

Established dissipative convergence up to the boundary

Abstract

We establish various criteria, which are known in the incompressible case, for the validity of the inviscid limit for the compressible Navier-Stokes flows considered in a general domain $\Omega$ in $\mathbb{R}^n$ with or without a boundary. In the presence of a boundary, a generalized Navier boundary condition for velocity is assumed, which in particular by convention includes the classical no-slip boundary conditions. In this general setting we extend the Kato criteria and show the convergence to a solution which is dissipative "up to the boundary". In the case of smooth solutions, the convergence is obtained in the relative energy norm.