Global Wellposeness for the 3D inhomogeneous incompressible Navier-Stokes equations

Abstract

This paper addresses the three-dimensional Navier-Stokes equations for an incompressible fluid whose density is permitted to be inhomogeneous. We establish a theorem of global existence and uniqueness of strong solutions for initial data with small $\dot{H}^{\frac12}$-norm, which also satisfies a natural compatibility condition. A key point of the theorem is that the initial density need not be strictly positive.