Infinite growth in vorticity gradient of compactly supported planar vorticity near Lamb dipole

Abstract

We prove linear in time filamentation for perturbations of the Lamb dipole, which is a traveling wave solution to the incompressible Euler equations in $\mathbb{R}^2$. The main ingredient is a recent nonlinear orbital stability result by Abe-Choi. As a consequence, we obtain linear in time growth for the vorticity gradient for all times, for certain smooth and compactly supported initial vorticity in $\mathbb{R}^2$. The construction carries over to some generalized SQG equations.