Stability and instability of Kelvin waves

Kyudong Choi, In-Jee Jeong

Published 2022-04-07Version 1

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 nonlinear stability results in the $L^1$ norm of the vorticity, for $m$-fold symmetric perturbations. This is obtained by proving that the Kelvin wave is a strict local maximizer of the energy functional in some admissible class of patches. Based on the $L^1$ stability, 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.