Global Well-Posedness of Inhomogeneous Navier–Stokes Equations with Bounded Density

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 $0\leq \rho _{0}\in L^\infty (\mathbb R^{2})$ is bounded away from zero and the initial velocity $u_{0}\in L^{2}(\mathbb R^{2})$. 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 $\dot H^{1/2}(\mathbb R^{3})$. The key difficulty is that one can only obtain $t^{1/2}\nabla u\in L^{2}(0,T; L^\infty (\mathbb R^{d}))$ instead of $\nabla u\in L^{1}(0, T; L^\infty (\mathbb R^{d}))$ when $u_{0}\in L^{2}(\mathbb R^{2})$ or $u_{0}\in \dot H^{1/2}(\mathbb R^{3})$, where the later estimate is crucial for the uniqueness of the solution in all known works.