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In the well-known book of Lions [{\em Mathematical topics in fluid mechanics.Incompressible models}, 1996], global existence results of finite energy weaksolutions 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 boundaryre\-gularity of density patches were listed as open problems. A breakthrough onLions' problems was recently made by Danchin and Mucha [The incompressibleNavier-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 ofPoincar\'{e}-type inequalities. In this paper, we obtain regularity anduniqueness of Lions' weak solutions for (INS) with \emph{only bounded andnonnegative initial density} and additional regularity only assumed for theinitial velocity, in the whole-space case $\mathbb R^d$, $d=2$ or $3$. Inparticular, our result allows us to study the evolution of a vacuum bubbleembedded in an incompressible fluid, as well as a patch of a homogeneous fluidembedded in the vacuum, which provides an answer to Lions' question in thewhole-space case.

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