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

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.

References in corpus (8)

• An extension problem related to the fractional Laplacian
• 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
• Invariant measures and global well posedness for SQG equation
• Stability of Lamb dipoles
• Traveling vortex pairs for 2D incompressible Euler equations