Asymptotic estimates for concentrated vortex pairs

TL;DR

This paper demonstrates that scaled maximizers of a vortex energy functional form a family of concentrated vortex pairs approaching point vortices with opposite signs, providing asymptotic estimates.

Contribution

It establishes the asymptotic behavior of vortex pairs as they concentrate and approach point vortices, extending previous existence and stability results.

Findings

Maximized vortex pairs become increasingly concentrated with scaling.

The scaled maximizers approach a pair of point vortices with equal magnitude and opposite signs.

A uniform bound for the support size of scaled maximizers is proved.

Abstract

In [Comm. Math. Phys. 324 (2013), 445--463], Burton-Lopes Filho-Nussenzveig Lopes studied the existence and stability of slowly traveling vortex pairs as maximizers of the kinetic energy penalized by the impulse relative to a prescribed rearrangement class. In this paper, we prove that after a suitable scaling transformation the maximization problem studied by Burton-Lopes Filho-Nussenzveig Lopes in fact gives rise to a family of concentrated vortex pairs approaching a pair of point vortices with equal magnitude and opposite signs. The key ingredient of the proof is to deduce a uniform bound for the size of the supports of the scaled maximizers.