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 $u_0\in H^s({\mathbb{R}}^2)$ for $s>0$ in 2-D, or $u_0\in H^1({\mathbb{R}}^3)$ satisfying $\mathrm{\mid }{u}_0{\mathrm{\mid }}_{L^2}\mathrm{\mid }\text{na}{u}_0{\mathrm{\mid }}_{L^2}$ being sufficiently small in 3-D. This in particular improves the most recent well-posedness result in [10], which requires the initial velocity $u_0\in H^2(\R^d)$ for the local well-posedness result, and a smallness condition on the fluctuation of the initial density for the global well-posedness result.