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

Abstract

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 \(\rho _0\) is discontinuous and the initial velocity \(u_0\) has critical regularity. Assuming that \(\rho _0\) is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity \(u_0\) belongs to the critical homogeneous Besov space \(\dot{B}^{-1+2/p}_{p,1}({\mathbb {R}}^{2})\) \((1<p<2)\) and, in the three-dimensional case, if \(u_0\) is small in \(\dot{B}^{-1+3/p}_{p,1}({\mathbb {R}}^{3})\ (1<p<3)\). 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 Zhang (Adv Math 363:107007, 2020) 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.