Strong solutions of the Navier–Stokes equations for isentropic compressible fluids

Abstract

We study strong solutions of the isentropic compressible Navier–Stokes equations in a domain Ω⊂ R 3 . We first prove the local existence of unique strong solutions provided that the initial data ρ 0 and u 0 satisfy a natural compatibility condition. The important point in this paper is that we allow the initial vacuum: the initial density may vanish in an open subset of Ω. We then prove a new uniqueness result and stability result. Our results are valid for unbounded domains as well as bounded ones.