Growth of perimeter for vortex patches in a bulk

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 slit out of an almost square patch. A similar result holds for an almost round patch with a thin handle in R-2. (c) 2020 Elsevier Ltd. All rights reserved.