Nonlinear stabilitty for steady vortex pairs

TL;DR

This paper proves the nonlinear orbital stability of steadily translating vortex pairs in 2D Euler flows using a variational approach that leverages invariance properties, extending previous existence results.

Contribution

It introduces a novel variational method to establish nonlinear stability of vortex pairs, applicable to a broad class of solutions.

Findings

Proves nonlinear orbital stability of vortex pairs.

Uses Kelvin's variational principle with invariance properties.

Applies to a wide class of vortex solutions.

Abstract

In this article, we prove nonlinear orbital stability for steadily translating vortex pairs, a family of nonlinear waves that are exact solutions of the incompressible, two-dimensional Euler equations. We use an adaptation of Kelvin's variational principle, maximizing kinetic energy penalised by a multiple of momentum among mirror-symmetric isovortical rearrangements. This formulation has the advantage that the functional to be maximized and the constraint set are both invariant under the flow of the time-dependent Euler equations, and this observation is used strongly in the analysis. Previous work on existence yields a wide class of examples to which our result applies.