Abstract In this paper, we prove the uniqueness of Lions’ weak solutions for the 2D inhomogeneous Navier–Stokes equations (INS) in the case when the initial density 0 ₀ L^ (R^2) is bounded away from zero and the initial velocity u₀ L^2 (R^2). We also extend a celebrated result by Fujita and Kato on the 3D incompressible Navier–Stokes equations to 3D (INS): the global well-posedness of 3D (INS) with bounded initial density and initial velocity being small in H^1/2 (R^3). The key difficulty is that one can only obtain t^1/2 u L^2 (0, T; L^ (R^d) ) instead of u L^1 (0, T; L^ (R^d) ) when u₀ L^2 (R^2) or u₀ H^1/2 (R^3), where the later estimate is crucial for the uniqueness of the solution in all known works.
Hao et al. (Mon,) studied this question.