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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 862186, 14 pages
http://dx.doi.org/10.1155/2011/862186
Research Article

Conservation of Total Vorticity for a 2D Stochastic Navier Stokes Equation

Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA

Received 19 July 2011; Revised 15 September 2011; Accepted 17 September 2011

Academic Editor: Rémi Léandre

Copyright © 2011 Peter M. Kotelenez and Bradley T. Seadler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider 𝑁 point vortices whose positions satisfy a stochastic ordinary differential equation on 2 𝑁 perturbed by spatially correlated Brownian noise. The associated signed point measure-valued empirical process turns out to be a weak solution to a stochastic Navier-Stokes equation (SNSE) with a state-dependent stochastic term. As the number of vortices tends to infinity, we obtain a smooth solution to the SNSE, and we prove the conservation of total vorticity in this continuum limit.