  In this paper, we solve Lions' open problem: {\it the uniqueness of weak
solutions for the 2-D inhomogeneous Navier-Stokes equations (INS)}. We first
prove the global existence of weak solutions to 2-D (INS) with bounded initial
density and initial velocity in $L^2(\mathbb R^2)$. Moreover, if the initial
density is bounded away from zero, then our weak solution equals to Lions' weak
solution, which in particular implies the uniqueness of Lions' weak solution.
We also extend a celebrated result by Fujita and Kato on the 3-D incompressible
Navier-Stokes equations to 3-D (INS): {\it the global well-posedness of 3-D
(INS) with bounded initial density and initial velocity being small in $\dot
H^{1/2}(\mathbb R^3)$}.
  The proof of the uniqueness is based on a surprising finding that the
estimate $t^{1/2}\nabla u\in L^2(0,T; L^\infty(\mathbb R^d))$ instead of
$\nabla u\in L^1(0, T; L^\infty(\mathbb R^d))$ is enough to ensure the
uniqueness of the solution.
