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.1 (c 0 )r x 2 , for 0 < r := x 2 1 + x 2 2 ≤ c 0 λ −1/2 , c 2 0 x 2 /λr 2 , for r > c 0 λ −1/2 ,

Infinite growth in vorticity gradient of compactly supported planar vorticity near Lamb dipole

“…(However, see [31,30,22] and references therein.) Existence and orbital stability of dipole structures are known for the entire range 0 < α < 1, thanks to the very recent work [3]. Based on the stability statement, we can prove the following result on gradient growth.…”

“…In this section, we illustrate the main steps in the proof of Proposition 1.1, which is largely parallel to that of Theorem 1.2, using the recent existence and stability theorems of Cao-Qin-Zhan-Zou [3].…”

“…• Existence of traveling waves. The authors in [3] prove existence of a smooth traveling wave solution θ…”

“…L to (1.5) for 0 < α < 1 in R 2 which is non-negative on R 2 + , odd symmetric with respect to the x 1 -axis, and supported precisely in the unit disc; see [3,Theorem 1.1]. Similarly as in [1], the existence statement is obtained by setting up a maximization problem involving conserved quantities for (1.5).…”

“…• Uniqueness and stability. In [3,Theorem 1.3], the authors demonstrate the orbital stability of θ (α)…”