Steady vortex patch solutions to the vortex-wave system

Abstract

In this paper we prove the existence of steady multiple vortex patch solutions to the vortex-wave system in a planar bounded domain. The construction is performed by solving a certain variational problem for the vorticity and studying its asymptotic behavior as the vorticity strength goes to infinity.where J(a, b) := (b, −a) denotes clockwise rotation through π 2 for any vector (a, b) ∈ R 2 , andJx |x| 2 is called the Biot-Savart kernel. The vorticity equation (1.2) means that the vorticity ω is transported by the divergence-free velocity field K * ω.Email addresses: dmcao@amt.ac.cn (Daomin Cao), wangguodong14@mails.ucas.ac.cn (Guodong Wang)

Steady vortex patches near a nontrivial irrotational flow

“…In this paper, we focus on the construction of steady vortex patches. There exist a great literatures dealing with this problem; see for example [8,9,10,13,20,21] and the references listed therein. An efficient method to study the vortex patch problem is the vorticity method.…”

Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability

“…When C = 0, such system was first introduced by Helmholtz [18,22] and proved rigorously in [26,37,28]. See also [32,9,13] and, for a related problem of the limit motion of concentrated vorticities in background vorticity distributions, see [27,20,3,10]. As the first step to understand the dynamics of the concentrated vorticities, through a perturbation approach, we shall study steady concentrated vorticities, both Lipschitz and piecewise constant on Ω, located near a non-degenerate critical configuration of H C with each vortical domain being an O(| r|r 2 j ) perturbation to the disk B r j (x * j ) of radius r j .…”

On the existence of vortex-wave systems to inviscid gSQG equation

“…However, as far as we know, few results are known for steady solutions to this system. The only two papers [10,11] for steady solutions of this system are available.…”

“…In [10], The author first studies the vortex wave system for Euler equation, which is the extension of the results in [9] to vortex wave systems. Then in [11], the author extended the results of [10] to the case of multiple vortices.…”

“…It is worth mentioning that the construction in [10,11] was based on the vorticity method, which was first established by Arnold [20] and further developed by many authors [4,6,5,21,22,8]. The advantage of the method in [10,11] is that solutions concentrating at a given strict local minimum point of the Kirchhoff-Routh function can be constructed. However, the vorticity method is no longer applicable for saddle point cases.…”