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

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 $d\ge 2$ and for general pressure laws.

Submitted

References in corpus (1)

• On the global regularity of 2-D density patch for inhomogeneous incompressible viscous flow

Cited by in corpus (1)

• Global unique solutions for the inhomogeneous Navier-Stokes equation with only bounded density, in critical regularity spaces