Stability and instability of Kelvin waves

Abstract

The m-waves of Kelvin are uniformly rotating patch solutions of the 2D Euler equations with m-fold rotational symmetry for m ≥ 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.

Stability of monotone vorticities in the half cylinder and infinite perimeter growth for patches

“…[6] showed a stability, up to translation, of the Hill's spherical vortex among non-negative axi-symmetric solutions in 3D. [9] presented an orbital stability in L 1 of m−fold Kelvin waves for symmetric perturbations and constructed an m−fold Kelvin wave which shows perimeter growth for sufficiently long finite time.…”

Uniqueness and stability of steady vortex rings

“…We refer the reader to [2,41] for some historical reviews of the achievements in experimental, analytical, and numerical studies of vortex rings. Though numerous contributions to stability of steady solutions to 2D Euler equations have been achieved, for which we refer the interested reader to [1,13,14,22,25,26,50] and references therein, the works on the stability of vortex rings seem to be relatively rare. Recently, Choi [24] established the orbital stability of Hill's vortex by a combination of the variational framework due to [32], the uniqueness result [4] and the concentrated compactness lemma of Lions [39].…”