OPTIMAL WELL-POSEDNESS FOR THE INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES SYSTEM WITH GENERAL VISCOSITY

In this paper we obtain new well-posedness results concerning a linear inhomogeneous Stokes-like system. These results are used to establish local well-posedness in the critical spaces for initial density rho(0) and velocity u(0) such that rho(0) - rho is an element of (B) over dot(p,1)(3/p) (R-3), u(0) is an element of (B) over dot(p,1)(3/p-1) (R-3), p is an element of (6/5,4) for the inhomogeneous incompressible Navier-Stokes system with variable viscosity. To the best of our knowledge, regarding the 3-dimensional case, this is the first result in a truly critical framework for which one does not assume any smallness condition on the density.