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