On Lions' density patch problem at a critical level of regularity

TL;DR

This paper proves global existence, uniqueness, and stability for a fluid density patch problem at critical regularity in 2D Navier--Stokes, showing the patch's Lipschitz regularity persists and exhibits rigid motion asymptotics.

Contribution

It establishes the preservation of Lipschitz regularity and long-term rigid motion behavior for density patches at critical regularity in 2D Navier--Stokes.

Findings

Global existence and uniqueness of the density patch solution.

Lipschitz regularity of the patch is preserved over time.

The patch's long-time dynamics tend to a rigid motion with an asymptotic domain.

Abstract

In this article, we study Lions' density patch problem in two space dimensions at critical regularity. We prove global existence, uniqueness, and stability for a fluid occupying a bounded Lipschitz region surrounded by vacuum and evolving according to the incompressible Navier--Stokes equations, with initial velocity in $\dot{B}^0_{2,1}(\mathbb{R}^2)$. Moreover, we show that the Lipschitz regularity of the patch is preserved, and that its long-time dynamics is a rigid motion leading to the emergence of an asymptotic domain.