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 ∈ H (R) for s > 0 in 2-D, or u0 ∈ H (R) satisfying ‖u0‖L2‖∇u0‖L2 being sufficiently small in 3-D. This in particular improves the most recent well-posedness result in [10], which requires the initial velocity u0 ∈ H (R) for the local well-posedness result, and a smallness condition on the fluctuation of the initial density for the global well-posedness result.