Free boundary regularity of vacuum states for incompressible viscous flows in unbounded domains

In the well-known book of Lions [{\em Mathematical topics in fluid mechanics. Incompressible models}, 1996], global existence results of finite energy weak solutions of the inhomogeneous incompressible Navier-Stokes equations (INS) were proved without assuming positive lower bounds on the initial density, hence allowing for vacuum. Uniqueness, regularity and persistence of boundary re\-gularity of density patches were listed as open problems. A breakthrough on Lions' problems was recently made by Danchin and Mucha [The incompressible Navier-Stokes equations in vacuum, {\em Comm. Pure Appl. Math.}, 72 (2019), 1351--1385] in the case where the fluid domain is either bounded or the torus. However, the case of unbounded domains was left open because of the lack of Poincar\'{e}-type inequalities. In this paper, we obtain regularity and uniqueness of Lions' weak solutions for (INS) with \emph{only bounded and nonnegative initial density} and additional regularity only assumed for the initial velocity, in the whole-space case $\mathbb R^d$, $d=2$ or $3$. In particular, our result allows us to study the evolution of a vacuum bubble embedded in an incompressible fluid, as well as a patch of a homogeneous fluid embedded in the vacuum, which provides an answer to Lions' question in the whole-space case.