Unique solvability for the density-dependent Navier–Stokes equations

Abstract

In this paper we consider the incompressible Navier–Stokes equations with a density-dependent viscosity in a bounded domain Ω of R n ( n = 2 , 3 ) . We prove the local existence of unique strong solutions for all initial data satisfying a natural compatibility condition. This condition is also necessary for a very general initial data. Moreover, we provide a blow-up criterion for the regularity of the strong solution. For these results, the initial density need not be strictly positive. It may vanish in an open subset of Ω .