Global Unique Solutions for the Inhomogeneous Navier-Stokes equations with only Bounded Density, in Critical Regularity Spaces

We here aim at proving the global existence and uniqueness of solutions to inhomogeneous incompressible Navier-Stokes system in case where initial density \(\rho _0\) is discontinuous velocity \(u_0\) has critical regularity. Assuming that close a positive constant, we obtain two-dimensional whenever belongs homogeneous Besov space \(\dot{B}^{-1+2/p}_{p,1}({\mathbb {R}}^{2})\) \((1<p<2)\) and, three-dimensional case, if small \(\dot{B}^{-1+3/p}_{p,1}({\mathbb {R}}^{3})\ (1<p<3)\). Next, still functional framework, establish statement valid large variations with, possibly, vacuum. Interestingly, our result implies Fujita-Kato type constructed by Zhang (Adv Math 363:107007, 2020) are unique. Our work relies on interpolation results, time weighted estimates maximal regularity Lorentz spaces (with respect variable) for evolutionary Stokes system.