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

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.