On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities (original) (raw)

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)) × (D 1 0 ∩ D 3)(Ω)) 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.