Finite time singularity formation for moving interface Euler equations

TL;DR

This paper introduces a new methodology to demonstrate finite-time singularity formation in moving interface problems governed by the incompressible Euler equations, including vortex sheets and rigid body interactions, revealing contrasting behaviors with previous models.

Contribution

It develops a general approach to prove finite-time singularities in Euler interface problems, contrasting with previous results where natural norms remained finite until contact.

Findings

Finite-time singularity in vortex sheets with surface tension.

Rigid body hits the bottom of the fluid domain in finite time.

Surface energy blows up at contact, with acceleration behavior depending on contact zone.

Abstract

This paper proposes a new general methodology for finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane. The first problem considered is the two-phase Euler 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 [4] and studied in a more general context in [8], for which the natural norm (in the one phase fluid) stays finite all the way until contact. The second problem considered involves the presence of a heavier rigid body moving in the perfect fluid. We first establish that the rigid body will hit the bottom of the fluid domain in finite time (in a more general context and very different methodology than first done for this problem in [19]). We next establish that a surface energy blows up, and characterize the acceleration of the rigid body at contact, depending on the nature of the contact zone: Acceleration opposes the motion at contact, and is either positive finite if the contact zone contains a curve, or infinite otherwise.