Local existence of solution to free boundary value problem for compressible Navier-Stokes equations

TL;DR

This paper proves the local-in-time existence of weak solutions for a free boundary problem involving multi-dimensional compressible Navier-Stokes equations with density-dependent viscosity, where the density vanishes at the free boundary.

Contribution

It establishes the first local existence result for weak solutions with density vanishing at the free boundary in multi-dimensional compressible Navier-Stokes equations with density-dependent viscosity.

Findings

Weak solutions exist locally in time

Density remains positive away from the free boundary

Solutions are regular except at the free boundary

Abstract

This paper is concerned with the free boundary value problem for multi-dimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. A local (in time) existence of weak solution is established, in particular, the density is positive and the solution is regular away from the free boundary.