Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation

TL;DR

This paper investigates slow traveling-wave solutions of the generalized surface quasi-geostrophic equation, establishing their existence, uniqueness, stability, and connection to vortex pair desingularization using variational methods.

Contribution

It introduces a new family of solutions for the gSQG equation and proves their stability and uniqueness, linking them to vortex pair desingularization.

Findings

Existence of a new family of global solutions.

Uniqueness of maximizers under variational framework.

Stability of solutions via concentration-compactness.

Abstract

In this paper, we systematically study the existence, asymptotic behaviors, uniqueness, and nonlinear orbital stability of traveling-wave solutions with small propagation speeds for the generalized surface quasi-geostrophic (gSQG) equation. Firstly we obtain the existence of a new family of global solutions via the variational method. Secondly we show the uniqueness of maximizers under our variational setting. Thirdly by using the variational framework, the uniqueness of maximizers and a concentration-compactness principle we establish some stability theorems. Moreover, after a suitable transformation, these solutions constitute the desingularization of traveling point vortex pairs.