Density-dependent incompressible viscous fluids in critical spaces

We study the unique solvability of density-dependent incompressible Navier-Stokes equations in the whole space R-N (N greater than or equal to 2). The celebrated results by Fujita and Kato devoted to the constant density case are generalized to the case when the initial density is close to a constant: we find local well posedness for large initial velocity, and global well posedness for initial velocity small with respect to the viscosity. Our functional setting is very close to the one used by Fujita and Kato.