Euler system with a polytropic equation of state as a vanishing viscosity limit

TL;DR

This paper demonstrates that smooth solutions of the Euler system with a polytropic equation of state can be obtained as limits of the Navier--Stokes--Fourier system with vanishing viscosity and heat conductivity, under various boundary conditions.

Contribution

It establishes the convergence of Navier--Stokes--Fourier solutions to Euler solutions in the vanishing viscosity limit, including cases with slip and no-slip boundary conditions.

Findings

Unconditional convergence under slip boundary conditions.

Extension to no-slip boundary conditions with additional boundary layer hypotheses.

Provides a rigorous link between viscous and inviscid gas dynamics models.

Abstract

We consider the Euler system of gas dynamics endowed with the incomplete equation of state relating the internal energy to the mass density and the pressure. We show that any sufficiently smooth solution can be recovered as a vanishing viscosity - heat conductivity limit of the Navier--Stokes--Fourier system with a properly defined temperature. The result is unconditional in the case of the Navier type (slip) boundary conditions and extends to the no-slip condition for the velocity under some extra hypotheses of Kato's type concerning the behavior of the fluid in the boundary layer.