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

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.

References in corpus (7)

• Desingularization of vortices for the Euler equation
• Existence of corotating and counter-rotating vortex pairs for active scalar equations
• Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations
• On the V-states for the generalized quasi-geostrophic equations
• Nonlinear stabilitty for steady vortex pairs
• Spectral stability of inviscid columnar vortices
• Invariant measures and global well posedness for SQG equation