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

We study strong solutions of the isentropic compressible Navier–Stokes equations in a domain Ω R3. We first prove the local existence of unique strong solutions provided that the initial data ρ0 and u0 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.