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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 ε 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 ε 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 (𝐑 2 ,dr) 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 ε0 and N (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:
60H15Stochastic partial differential equations
76D05Navier-Stokes equations (fluid dynamics)
60F99Limit theorems (probability)
35K55Nonlinear parabolic equations
35A35Theoretical approximation to solutions of PDE
65M99Numerical methods for IVP of PDE
65N99Numerical methods for BVP of PDE