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

In this paper, we shall prove the global existence of weak solutions to 3D inhomogeneous incompressible Navier-Stokes system $({\rm INS})$ 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, $\dot B^{\frac{1}{2}}_{2,1}.$ 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 \emph{critical functional framework} to $({\rm INS})$ 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 $({\rm INS}).$