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Vortex theory approach to stochastic hydrodynamics. (English) Zbl 1130.76030
Summary: We study a jump-diffusion type vorticity model, describing evolution of an incompressible homogeneous viscous fluid in 2 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 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. We also present a continuous diffusion approximation to the above vorticity model.
MSC:
76D06Statistical solutions of Navier-Stokes and related equations
76D17Viscous vortex flows
76M35Stochastic analysis (fluid mechanics)
60H15Stochastic partial differential equations