Global Unique Solvability of Inhomogeneous Navier-Stokes Equations with Bounded Density

In this paper, we prove the global existence and uniqueness of solution to d-dimensional (for d = 2, 3) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from above and below by some positive constants, and with initial velocity u0 ∈ Hs( R 2) for s > 0 in 2-D, or u0 ∈ H1( R 3) satisfying ‖u0‖L2 ‖∇u0‖L2 being sufficiently small in 3-D. This in particular improves the most recent well-posedness result in [10 Danchin , R. , Mucha , P.B. ( 2013 ). Incompressible flows with piecewise constant density . Arch. Rat. Mech. Anal. 207 : 991 – 1023 .[Crossref], [Web of Science ®] , [Google Scholar]], which requires the initial velocity u0 ∈ H2( R d) for the local well-posedness result, and a smallness condition on the fluctuation of the initial density for the global well-posedness result.