Stability of vortex quadrupoles with odd-odd symmetry

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.