Optimal well-posedness for the inhomogeneous incompressible Navier-Stokes system with general viscosity

In this paper we obtain new well-possedness results concerning a linear inhomogenous Stokes-like system. These results are used to establish local well-posedness in the critical spaces for initial density ρ0 and velocity u0 such that ρ0−ρ ∈ Ḃ 3 p p,1(R ), u0 ∈ Ḃ 3 p −1 p,1 (R ), p ∈ ( 6 5 , 4 ) , for the inhomogeneous incompressible Navier-Stokes system with variable viscosity. To the best of our knowledge, regarding the 3D case, this is the rst result in a truly critical framework for which one does not assume any smallness condition on the density.