Relaxation Theory for Point Vortices

Abstract

We study relaxation dynamics of the mean field of many point vortices from quasiequilibrium to equilibrium. Maximum entropy production principle implies four consistent equations concerning relaxation-equilibrium states and patch-point vortex models. Point vortex relaxation equation coincides with Brownian point vortex equation in micro-canonical setting. Mathematical analysis to point vortex relaxation equation is done in accordance with the Smoluchowski-Poisson equation.

Relaxation Dynamics of Point Vortices

“…by ( 30) and (43). This system is the stationary state of ( 1) and (2) introduced by [4]. The first two equalities…”

“…In more details, we take a cut-off function, denoted by ψ, satisfying (121), using a local conformal mapping, and then put φ ¼ ψ 4 .…”

“…It was derived, first, for Brownian point vortices by [1,2], with β ¼ β t ðÞstanding for the inverse temperature. Then, [3,4] reached it by the Lynden-Bell theory [5] of relaxation dynamics, that is, as a model describing the movement of the mean field of many point vortices, from quasi-stationary state to the stationary state. This model is consistent to the Onsager theory [6][7][8][9][10][11][12] on stationary states and also the patch model proposed by [13,14], that is,…”

“…Henceforth, C > 0 stands for a generic constant. In the previous work [4] we studied radially symmetric solutions and obtained a criterion for the existence of the solution global in time. Here, we refine the result as follows, where B 0, 1 ðÞ denotes the unit ball.…”