Stability of vortex quadrupoles with odd-odd symmetry

TL;DR

This paper proves the global-in-time stability of vortex quadrupoles with odd symmetry in 2D Euler equations, overcoming energy maximization issues by using dynamical analysis and new estimates.

Contribution

It introduces a novel approach combining dynamical information and pointwise estimates to establish vortex stability without relying on energy maximization solutions.

Findings

Vortex quadrupoles with odd symmetry are globally stable over time.

The method applies to stability of Lamb dipoles moving apart.

New pointwise Biot--Savart kernel estimates are developed.

Abstract

For the 2D incompressible Euler equations, we establish global-in-time ($t \in \mathbb{R}$) stability of vortex quadrupoles satisfying odd symmetry with respect to both axes. Specifically, if the vorticity restricted to a quadrant is signed, sufficiently concentrated and close to its radial rearrangement up to a translation in $L^1$, we prove that it remains so for all times. The main difficulty is that the kinetic energy maximization problem in a quadrant -- the typical approach for establishing vortex stability -- lacks a solution, as the kinetic energy continues to increase when the vorticity escapes to infinity. We overcome this by taking dynamical information into account: finite-time desingularization result is combined with monotonicity of the first moment and a careful analysis of the interaction energies between vortices. The latter is achieved by new pointwise estimates on the Biot--Savart kernel and quantitative stability results for general interaction kernels. Moreover, with a similar strategy we obtain stability of a pair of opposite-signed Lamb dipoles moving away from each other.