On the Long Time Behaviour of Stochastic Vortices Systems
2014, v.20, Issue 4, 675-704
In this paper, we are interested in the long-time behaviour of stochastic systems of $n$ interacting vortices: the position in $R^2$ of each vortex evolves according to a Brownian motion and a drift summing the influences of the other vortices computed through the Biot and Savart kernel and multiplied by their respective vorticities. For fixed n, we perform the rescalings of time and space used successfully by Gallay and Wayne [T. Gallay and C.E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Commun. Math. Phys., 2005, v. 255, N 1, 97-129] to study the long-time behaviour of the vorticity formulation of the two dimensional incompressible Navier - Stokes equation, which is the limit as $n\to\infty$ of the weighted empirical measure of the system under mean-field interaction. When all the vorticities share the same sign, the 2n-dimensional process of the rescaled positions of the vortices is shown to converge exponentially fast as time goes to infinity to some invariant measure which turns out to be Gaussian if all the vorticities are equal. In the particular case n=2 of two vortices, we prove exponential convergence in law of the 4-dimensional process to an explicit random variable, whatever the choice of the two vorticities. We show that this limit law is not Gaussian when the two vorticities are not equal.
Keywords: vortices,stochastic differential equation,long-time behavior,Lyapunov function,logarithmic Sobolev inequality