On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations

TL;DR

This paper constructs global strong solutions for the 2-D inhomogeneous Navier-Stokes equations with minimal initial data assumptions, resolving the density patch problem and showing the equivalence of weak and strong solutions under certain conditions.

Contribution

It provides the first comprehensive proof that Lions' weak solutions coincide with strong solutions for density patches in 2-D Navier-Stokes equations.

Findings

Global strong solutions exist under minimal initial conditions.

Lions' weak solutions are equivalent to strong solutions for density patches.

The regularity of the initial density domain is preserved over time.

Abstract

In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubbles, we prove that Lions' s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions: {\it for the density patch data $\rho_0=1_{D}$ with a smooth bounded domain $D\subset\mathbb{R}^2$, the regularity of $D$ is preserved by the time evolution of Lions's weak solution.}