On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R ³. We first prove the local existence of solutions (ρ,u) in C([0,T_*]; (ρ ^∞ +H ³(Ω)) × [InlineMediaObject not available: see fulltext.] under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T _**)×Ω for some T _** (0,T _*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.