On steady solutions to vacuumless Newtonian models of compressible flow

TL;DR

This paper proves the existence of weak solutions for steady compressible Navier-Stokes equations with vacuum, showing that under certain conditions, density remains bounded away from zero and solutions exhibit regularity.

Contribution

It introduces a method to establish weak solutions with bounded density for vacuum-including models, extending classical regularity results.

Findings

Existence of weak solutions with bounded density in vacuum conditions

Density and velocity gradient are at least Hölder continuous

Solutions are constructed in a bounded domain with slip boundary conditions

Abstract

We prove the existence of weak solutions to the steady compressible Navier-Stokes system in the barotropic case for a class of pressure laws singular at vacuum. We consider the problem in a bounded domain in R^2 with slip boundary conditions. Due to appropriate construction of approximate solutions used in proof, obtained density is bounded away from 0 (and infinity). Owing to a classical result by P.-L. Lions, this implies that density and gradient of velocity are at least H\"older continuous, which does not generally hold for the classical isentropic model in the presence of vacuum.