Local and global well-posedness results for flows of inhomogeneous viscous fluids

Abstract

This paper is devoted to the the study of density-dependent, incompressible Navier-Stokes equations with periodic boundary conditions, or in the whole space. We aim at stating well-posedness in functional spaces as close as possible to the ones imposed by the scaling of the equations. Preliminary results have been obtained in [5] under the assumption that the density is close to a constant. Getting rid of this assumption (by allowing smoother data if necessary) is the main motivation of the present paper. Local well-posedness is stated for data (ρ0, u0) such that (ρ0 - cste) ∈ H N/2 +α and inf ρ0 > 0, and u0 ∈ H N/2 -1+β. The indices α, β > 0 may be taken arbitrarily small. We further derive a blow-up criterion which entails global well-posedness in dimension N = 2 if there is no vacuum initially.