On classical solutions of the compressible NavierStokes equations with nonnegative initial densities
Preprint Series # 676 Cho, Yonggeun and Kim, Hyunseok On classical solutions of the compressible NavierStokes equations with nonnegative initial densities. (2004); Abstract We study the NavierStokes 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 $\Omega$.
