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A stochastic Navier-Stokes equation for the vorticity of a two-dimensional fluid. (English) Zbl 0851.60062
Summary: The Navier-Stokes equation for the vorticity of a viscous and incompressible fluid in ${𝐑}^{2}$ is analyzed as a macroscopic equation for an underlying microscopic model of randomly moving vortices. We consider $N$ point vortices whose positions satisfy a stochastic ordinary differential equation on ${𝐑}^{2N}$, where the fluctuation forces are state dependent and driven by Brownian sheets. The state dependence is modeled to yield a short correlation length $\epsilon$ between the fluctuation forces of different vortices. The associated signed point measure-valued empirical process turns out to be a weak solution to a stochastic Navier-Stokes equation (SNSE) whose stochastic term is state dependent and small if $\epsilon$ is small. Thereby we generalize the well known approach to the Euler equation to the viscous case. The solution is extended to a large class of signed measures conserving the total positive and negative vorticities, and it is shown to be a weak solution of the SNSE. For initial conditions in ${L}_{2}\left({𝐑}^{2},dr\right)$ the solutions are shown to live on the same space with continuous sample paths and an equation for the square of the ${L}_{2}$-norm is derived. Finally we obtain the macroscopic NSE as the correlation length $\epsilon \to 0$ and $N\to \infty$ (macroscopic limit), where we assume that the initial conditions are sums of $N$ point measures. As a corollary to the above results we obtain the solution to a bilinear stochastic partial differential equation which can be interpreted as the temperature field in a stochastic flow.
##### MSC:
 60H15 Stochastic partial differential equations 76D05 Navier-Stokes equations (fluid dynamics) 60F99 Limit theorems (probability) 35K55 Nonlinear parabolic equations 35A35 Theoretical approximation to solutions of PDE 65M99 Numerical methods for IVP of PDE 65N99 Numerical methods for BVP of PDE