Cavity collapse and kinetic energy concentration in three-dimensional free-boundary Euler flow

Abstract

We construct smooth axisymmetric solutions of the three-dimensional incompressible free-boundary Euler equations in bounded domains for which a tubular cavity collapses onto a circle at a finite time $T$. The velocity has a single-valued potential, and the construction applies without surface tension and with any fixed positive surface tension. Both boundary components remain embedded and disjoint before collapse, while the outer boundary converges smoothly. The terminal outer meridional domain may be any smooth bounded simply connected domain whose closure is separated from the rotation axis; the interior point generating the collapse circle and the positive concentrated energy may be prescribed independently. We prove a sharp collapse law with radius asymptotic to an explicit multiple of $(T-t)^{1/2}|\log(T-t)|^{-1/4}$, determine curvature blow-up asymptotics, and show that kinetic energy concentrates uniformly on the circle. For a canonical reflection symmetric family, we identify a common cylindrical spacetime limit and sharp velocity and pressure profiles. At every fixed spatial point in the region enclosed by the limiting outer boundary and away from the collapse circle, the velocity tends to zero while the pressure tends to positive infinity. For this family, we characterize the topology of the limiting material map and prove its optimal $1/2$-Hölder regularity and sharp $W^{1,p}$ threshold $1\le p<2$, with first derivatives in weak $L^2$ but not in $L^2$. These canonical solutions admit weak continuations that preserve mass and momentum and whose total energy drops at collapse by exactly the concentrated kinetic energy.