Stationary flows for compressible viscous fluid in a perturbed half-space

TL;DR

This paper proves the unique existence and asymptotic stability of multidirectional stationary solutions for the compressible Navier--Stokes equations in a perturbed half-space, extending known results from the classical half-space case.

Contribution

It establishes the existence and stability of stationary solutions that depend on all directions in a perturbed half-space, unlike the classical case.

Findings

Existence of multidirectional stationary solutions

Asymptotic stability of these solutions

Extension from classical to perturbed half-space

Abstract

We consider the compressible Navier--Stokes equation in a perturbed half-space with an outflow boundary condition as well as the supersonic condition. For a half-space, it has been known that a certain planar stationary solution exist and it is time-asymptotically stable. The planar stationary solution is independent of the tangential directions and its velocities of the tangential directions are zero. In this paper, we show the unique existence of stationary solutions for the perturbed half-space. The feature of our work is that our stationary solution depends on all directions and has multidirectional flow. Furthermore, we also prove the asymptotic stability of this stationary solution.