Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in 3-D bounded domain

TL;DR

This paper establishes uniform regularity and convergence of solutions from the full compressible Navier-Stokes system to the Euler system in a 3-D bounded domain, considering vanishing viscosity and heat conductivity.

Contribution

It proves the existence of a unique strong solution with uniform bounds and demonstrates the vanishing dissipation limit in a bounded domain with Navier-slip boundary conditions.

Findings

Existence of a unique strong solution independent of viscosity and heat conductivity.

Uniform bounds in Sobolev spaces for the solution.

Convergence rate of Navier-Stokes solutions to Euler solutions.

Abstract

In the present paper, we study the uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system whose viscosity and heat conductivity are allowed to vanish at different order. The problem is studied in a 3-D bounded domain with Navier-slip type boundary conditions \eqref{1.9}. It is shown that there exists a unique strong solution to the full compressible Navier-Stokes system with the boundary conditions \eqref{1.9} in a finite time interval which is independent of the viscosity and heat conductivity. The solution is uniform bounded in $W^{1,\infty}$ and a conormal Sobolev space. Based on such uniform estimates, we prove the convergence of the solutions of the full compressible Navier-Stokes to the corresponding solutions of the full compressible Euler system in $L^\infty(0,T;L^2)$,$L^\infty(0,T;H^1)$ and $L^\infty([0,T]\times\Omega)$ with a rate of convergence.