Finite time singularity formation for moving interface Euler equations

Daniel Coutand

Published 2017-01-06Version 1

This paper establishes finite in time singularity formation for two types of moving boundary problems involving the incompressible and irrotational Euler equations in the plane. The first problem considered involves the presence of a heavier rigid body moving in the perfect fluid for which well-posedness was recently established in [15]. We first establish that the rigid body will hit the bottom of the fluid domain in finite time, therefore forming a cusp singularity in the fluid domain, which is problematic for elliptic estimates. Next are established a set of strange and deep properties satisfied by the fluid velocity and pressure fields at the time of contact. The second problem considered is the two-phases vortex sheets problem with surface tension for which is proved the finite time singularity of the natural norm of the problem for suitable initial data. This is in striking contrast with the case of finite time splash and splat singularity formation for the one phase Euler equations introduced in [3] and studied in a more general context in [5], for which the natural norm (in the one phase fluid) stays finite all the way until contact.