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

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. 3 More precisely, given initial density such that a0 (sic) 1/rho 0-1 is an element of B-p,1(3/p)(R-3) and the initial velocity u(0) = (u(0)(h), u(0)(3)) is an element of Bp-1 (-1+2/p,1/p) (R-3), with u(0)(h) belonging to H-1(R-3), then the inhomogeneous NS system has a unique global solution provided that (parallel to a(0)parallel to(3/p)(Bp,1) + parallel to Lambda(-1)(h) partial derivative(3)u(0)parallel to(Bp-1-1+2/p,1/p) being sufficiently small for some bounded functionf depending on parallel to u(0)parallel to(-1+2/p,1/p)(p,1) and parallel to u(0)(h)parallel to (H1). 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).