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In this paper, we shall prove the global existence of weak solutions to 3Dinhomogeneous incompressible Navier-Stokes system $({\rm INS})$ with initialdensity in the bounded function space and having a positive lower bound andwith initial velocity being sufficiently small in the critical Besov space,$\dot B^{\f12}_{2,1}.$ This result corresponds to the Fujita-Kato solutions ofthe classical Navier-Stokes system. The same idea can be used to prove theglobal existence of weak solutions in the \emph{critical functional framework}to $({\rm INS})$ with one component of the initial velocity being large and canalso be applied to provide a lower bound for the lifespan of smooth enoughsolutions of $({\rm INS}).$

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