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 R2. 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 R2. The construction carries over to some generalized SQG equations.