Growth of perimeter for vortex patches in a bulk

TL;DR

This paper constructs specific vortex patches in 2D Euler flows on torus and plane with initially bounded perimeter that can grow arbitrarily large in finite time, demonstrating dynamic boundary evolution.

Contribution

It introduces explicit constructions of vortex patches with smooth boundaries exhibiting finite-time perimeter growth in 2D Euler equations.

Findings

Perimeter of vortex patches can grow arbitrarily large in finite time.

Constructed patches include a thin handle or stick, affecting boundary length.

Results hold for both torus and Euclidean plane settings.

Abstract

We consider the two-dimensional incompressible Euler equations. We construct vortex patches with smooth boundary on $T^2$ and $R^2$ whose perimeter grows with time. More precisely, for any constant $M > 0$, we construct a vortex patch in $T^2$ whose smooth boundary has length of order 1 at the initial time such that the perimeter grows up to the given constant $M$ within finite time. The construction is done by cutting a thin stick out of an almost square patch. A similar result holds for an almost round patch with a thin handle in $R^2$.