Planar vortex patch problem in incompressible steady flow

Steady vortex patch solutions to the vortex-wave system

“…Our purpose in the present paper is to extend the result in [13] to the vortex-wave system. To be more precise, we prove that for any given strict local minimum point of the Kirchhoff-Routh function, there exists a family of steady vortex patch solutions to the vortex-wave system that shrinks to this point.…”

“…According to the point vortex model, for each point vortex concentrated at x i , the velocity it produces is κ i K(· − x i ), and it moves by the velocity field produced by all the other N − 1 point vortices(not included itself). The asymptotic nature of the point vortex approximation has been analyzed in many papers, see [13,23,27,30] for example. Now it is natural to consider the mixed problem in which the vorticity consists of a continuously distributed part and N concentrated vortices.…”

“…In this case the Kirchhoff-Routh function plays an essential role. On the one hand, any critical point of the Kirchhoff-Routh function is a stationary point of the vortex model, and it has been shown in [13] that if this critical point is non-degenerate, then there exists a family of steady vortex patch solutions of the Euler equation shrinking to that critical point. Similar results can also be found in [12,27,28,31].…”

Steady vortex patches near a nontrivial irrotational flow

“…Later Burton [5,6] considered the maximization of the kinetic energy on general rearrangement classes, and by which he found more dynamically possible equilibria of planar vortex flows. Another method to study the vortex patch problem is developed by Cao et al [8]. By using a reduction argument for the stream function, they obtained steady multiple vortex patches near any given nondegenerate critical point of the Kirchhoff-Routh function.…”

“…Most of the previous results in [8,13,20,21] were concerned with the desingularization of point vortices. According to the vortex model, the evolution of a finite number of concentrated vortices in two dimensions is described by a dynamical system involving the Kirchhoff-Routh function; See [15] for a general discussion.…”