A well-posedness result for viscous compressible fluids with only bounded density

TL;DR

This paper proves local existence and uniqueness of solutions for viscous compressible fluids with only bounded density, extending previous results to more general conditions and dimensions.

Contribution

It establishes well-posedness for solutions with bounded density and introduces new uniqueness results under specific regularity assumptions.

Findings

Existence of local-in-time solutions with bounded density.

Uniqueness when density has striated regularity or across smooth interfaces.

Results valid in any dimension d ≥ 2 for general pressure laws.

Abstract

We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in the $L^\infty$ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension $d\geq2$ and for general pressure laws.