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

Abstract

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.

23 pages