Global unique solutions for the inhomogeneous Navier-Stokes equation with only bounded density, in critical regularity spaces

We here aim at proving the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system in the case where the initial density is discontinuous and the initial velocity has critical regularity. Assuming that the initial density is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity belongs to some critical homogeneous Besov space (and in small in the three-dimensional case). Next, still in a critical functional framework, we establish a uniqueness statement that is valid in the case of large variations of density with, possibly, vacuum. Interestingly, our result implies that the Fujita-Kato type solutions constructed by P. Zhang in are unique. Our work relies on interpolation results, time weighted estimates and maximal regularity estimates in Lorentz spaces (with respect to the time variable) for the evolutionary Stokes system.