On local strong solutions to the three-dimensional nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum

Abstract

We study the three-dimensional nonhomogeneous Navier–Stokes equations with density-dependent viscosity and vacuum on Ω⊂R3, which is either a bounded domain or a usual unbounded one such as the whole space R3 and an exterior one. In particular, the initial density can have compact support when Ω is unbounded. For initial data without additional compatibility conditions, we prove that there exists a unique local strong solution to the initial and initial boundary value problems. Moreover, the continuous of strong solutions on the initial data is derived under an additional compatibility condition.