A Well-Posedness Result for Viscous Compressible Fluids with Only Bounded Density
MFO Scientific ProgramOWLF 2017
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  with a uniqueness statement, and is valid in any dimension $d\geq2$ and for general pressure laws.