Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density

In this paper, we solve Lions' open problem: { 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 L²( R²). 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): { the global well-posedness of 3-D (INS) with bounded initial density and initial velocity being small in H1/2( R³)}. The proof of the uniqueness is based on a surprising finding that the estimate t1/2∇ u∈ L²(0,T; L^∞( Rᵈ)) instead of ∇ u∈ L¹(0, T; L^∞( Rᵈ)) is enough to ensure the uniqueness of the solution.