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