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

TL;DR

This paper demonstrates linear in time growth of the vorticity gradient near the Lamb dipole in 2D Euler equations, extending to some generalized SQG equations, based on stability analysis.

Contribution

It establishes linear in time filamentation for perturbations of the Lamb dipole and extends the result to certain generalized SQG equations.

Findings

Linear in time growth of vorticity gradient near Lamb dipole

Extension of results to some generalized SQG equations

Utilization of nonlinear orbital stability for analysis

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.