Unique solvability of the initial boundary value problems for compressible viscous fluids

Abstract

We study the Navier–Stokes equations for compressible barotropic fluids in a domain Ω⊂ R 3 . We first prove the local existence of the unique strong solution, provided the initial data satisfy a natural compatibility condition. The initial density needs not be bounded away from zero; it may vanish in an open subset ( vacuum) of Ω or decay at infinity when Ω is unbounded. We also prove a blow-up criterion for the local strong solution, which is new even for the case of positive initial densities. Finally, we prove that if the initial vacuum is not so irregular, then the compatibility condition of the initial data is necessary and sufficient to guarantee the existence of a unique strong solution.