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November, 1995 A Stochastic Navier-Stokes Equation for the Vorticity of a Two-Dimensional Fluid
Peter Kotelenez
Ann. Appl. Probab. 5(4): 1126-1160 (November, 1995). DOI: 10.1214/aoap/1177004609


The Navier-Stokes equation for the vorticity of a viscous and incompressible fluid in $\mathbf{R}^2$ 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 $\mathbf{R}^{2N}$, where the fluctuation forces are state dependent and driven by Brownian sheets. The state dependence is modeled to yield a short correlation length $\varepsilon$ 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 $\varepsilon$ 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 $L_2(\mathbf{R}^2, dr)$ 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 $\varepsilon \rightarrow 0$ and $N \rightarrow \infty$ (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.


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Peter Kotelenez. "A Stochastic Navier-Stokes Equation for the Vorticity of a Two-Dimensional Fluid." Ann. Appl. Probab. 5 (4) 1126 - 1160, November, 1995.


Published: November, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0851.60062
MathSciNet: MR1384369
Digital Object Identifier: 10.1214/aoap/1177004609

Primary: 60H15
Secondary: 35A35 , 35A40 , 35K55 , 60F99 , 76D05

Keywords: eddy diffusion , macroscopic limit , Navier-Stokes equation , random vortices , Stochastic partial differential equation , stochastic temperature field , viscous diffusion

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.5 • No. 4 • November, 1995
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