Stability and instability of Kelvin waves

TL;DR

This paper proves the nonlinear stability of Kelvin waves close to a disc in the 2D Euler equations, showing long-term stability and filamentation phenomena, and extends the analysis to annular patches.

Contribution

It establishes the first unconditional nonlinear stability result for Kelvin waves near a disc, using a variational approach and Lagrangian bootstrap techniques.

Findings

Kelvin waves are strict local energy maximizers among symmetric patches.

Long-time filamentation occurs near Kelvin waves, consistent with numerical observations.

Unconditional stability is achieved without support conditions on perturbation evolution.

Abstract

The $m$-waves of Kelvin are uniformly rotating patch solutions of the 2D Euler equations with $m$-fold rotational symmetry for $m\geq 2$. For Kelvin waves sufficiently close to the disc, we prove a nonlinear stability result up to an arbitrarily long time in the $L^1$ norm of the vorticity, for $m$-fold symmetric perturbations. To obtain this result, we first prove that the Kelvin wave is a strict local maximizer of the energy functional in some admissible class of patches, which had been claimed by Wan in 1986. This gives an orbital stability result with a support condition on the evolution of perturbations, but using a Lagrangian bootstrap argument which traces the particle trajectories of the perturbation, we are able to drop the condition on the evolution. Based on this unconditional stability result, we establish that long time filamentation, or formation of long arms, occurs near the Kelvin waves, which have been observed in various numerical simulations. Additionally, we discuss stability of annular patches in the same variational framework.