Global Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes system

Abstract

In this paper, we shall prove the global existence of weak solutions to 3D inhomogeneous incompressible Navier-Stokes system (INS) with initial density in the bounded function space and having a positive lower bound and with initial velocity being sufficiently small in the critical Besov space, B˙2,112. This result corresponds to the Fujita-Kato solutions of the classical Navier-Stokes system. The same idea can be used to prove the global existence of weak solutions in the critical functional framework to (INS) with one component of the initial velocity being large and can also be applied to provide a lower bound for the lifespan of smooth enough solutions of (INS).