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

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 ${\dot{B}}_{2,1}^0({\mathbb{R}}^2)$. 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.