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 .