Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation

TL;DR

This paper proves the asymptotic stability of stationary solutions to the full compressible Navier-Stokes equations with large initial disturbances in a half-line domain, under small boundary perturbations and no restriction on the adiabatic exponent.

Contribution

It establishes the stability of stationary solutions for the full compressible Navier-Stokes equations with large initial perturbations without restrictions on the adiabatic exponent, using energy methods.

Findings

Stationary solutions are asymptotically stable under large initial perturbations.

Positive uniform bounds for density and temperature are obtained over time and space.

Stability holds with small boundary strength regardless of the adiabatic exponent.

Abstract

We investigate the large-time behavior of solutions to an outflow problem of the full compressible Navier-Stokes equations in the half line. The non-degenerate stationary solution is shown to be asymptotically stable under large initial perturbation with no restriction on the adiabatic exponent $\gamma$, provided that the boundary strength is sufficiently small. The proofs are based on the standard energy method and the crucial step is to obtain positive lower and upper bounds of the density and the temperature uniformly in time and space.