Summary: Mathematical and Computer Modelling 45 (2007) 13191341
Vortex theory approach to stochastic hydrodynamics
Department of Statistics, The University of Michigan, 439 West Hall, 1085 S. University Ave., Ann Arbor, MI 48109-1107, USA
Received 5 June 2006; accepted 9 November 2006
The objective of the paper is to study a jump-diffusion type vorticity model, describing evolution of an incompressible
homogeneous viscous fluid in R2 in terms of its rotation. The model arises from a particle systems perspective, adopted in the
point vortex theory, and represents a measure-valued stochastic partial differential equation (SPDE) whose solution, under certain
conditions, is an empirical process generated by a finite system of randomly moving vortices, which interact via a (regularized)
logarithmic potential and are driven by suitable independent space-time Wiener processes and compensated Poisson random
measure. A continuous diffusion approximation to the above vorticity model is also presented.
c 2006 Elsevier Ltd. All rights reserved.
Keywords: Measure-valued stochastic partial differential equation; Stochastic NavierStokes equation; Vorticity; Jump-diffusion
Turbulence has long intrigued physicists and mathematicians alike. While being a widely observed and an
extremely important phenomenon for engineering and environmental applications related to atmospheric and oceanic
sciences, it presents such an abundance of open mathematical problems and experimental and numerical challenges,
which few (if any) other natural phenomena can really match. The mathematical history of fluid dynamics began with