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

TL;DR

This paper proves the global existence and uniqueness of weak solutions for 2-D inhomogeneous Navier-Stokes equations with bounded density, and extends well-posedness results to 3-D under small initial data conditions.

Contribution

It solves Lions' open problem on the uniqueness of weak solutions for 2-D INS and extends Fujita-Kato's 3-D results to inhomogeneous cases with bounded initial density.

Findings

Proved global existence of weak solutions in 2-D with bounded initial density.

Established uniqueness when initial density is bounded away from zero.

Extended 3-D well-posedness results to inhomogeneous Navier-Stokes with small initial data.

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.