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

Abstract

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain $\Omega$ of $\mathbf{R}^3$. We first prove the local existence of solutions $(\rho, u)$ in $C([0,T_* ]; ( \rho^\infty + H^3 (\Omega ) ) \times ( D_0^1 \cap D^3 )(\Omega ) )$ 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 $(\rho , u)$ is a classical solution in $(0,T_{**}) \times \Omega$ for some $T_{**} \in (0, T_* ]$. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset ({\it vacuum}) of $\Omega$