Partial regularity of the 3-D inhomogeneous Navier-Stokes equations

Abstract

In this paper, we first introduce the concept of suitable weak solutions for the inhomogeneous Navier-Stokes equations. In the case when the initial density is close to a positive constant, by combining local energy inequality, Sobolev embedding, pressure estimate and blow-up analysis, we establish interior regularity criterion of suitable weak solutions. Finally, we show that the 1-D Hausdorff measure of the set of possible singular points of suitable weak solutions vanishes by applying the interior regularity criterion.