Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation

TL;DR

This paper constructs and proves the stability of smooth, circular, counter-rotating vortex pairs for the generalized surface quasi-geostrophic equation, extending classical vortex solutions.

Contribution

It introduces a variational method to construct and analyze the stability of vortex pairs with circular supports for the gSQG equation, analogous to Lamb dipoles.

Findings

Existence of smooth traveling vortex pairs with circular supports.

Uniqueness and compactness of maximizers in the variational framework.

Orbital stability of the constructed vortex pairs.

Abstract

In this paper, we construct smooth travelling counter-rotating vortex pairs with circular supports for the generalized surface quasi-geostrophic equation. These vortex pairs are analogues of the Lamb dipoles for the two-dimensional incompressible Euler equation. The solutions are obtained by maximization of the energy over some appropriate classes of admissible functions. We establish the uniqueness of maximizers and compactness of maximizing sequences in our variational setting. Using these facts, we further prove the orbital stability of the circular vortex pairs for the gSQG equation.