Global well-posedness of 3D inhomogeneous Navier–Stokes system with small unidirectional derivative

Abstract

In Liu and Zhang (2020 Arch. Ration. Mech. Anal. 235 1405–44); Liu et al (2020 Arch. Ration. Mech. Anal. 238 805–43), the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the (anisotropic) Navier–Stokes (NS) system has a unique global solution. The goal of this paper is to extend this type of result to the 3D inhomogeneous (density-dependent) NS system. More precisely, given initial density such that and the initial velocity with belonging to , then the inhomogeneous NS system has a unique global solution provided that being sufficiently small for some bounded function f depending on and . This provide a more general result that of Chemin et al (2014 Commun. Math. Phys. 272 529–66); Chemin and Zhang (2015 Commun. PDE 40 878–96).