## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 5, Number 4 (1995), 1126-1160.

### A Stochastic Navier-Stokes Equation for the Vorticity of a Two-Dimensional Fluid

#### Abstract

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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 5, Number 4 (1995), 1126-1160.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177004609

**Digital Object Identifier**

doi:10.1214/aoap/1177004609

**Mathematical Reviews number (MathSciNet)**

MR1384369

**Zentralblatt MATH identifier**

0851.60062

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60]

Secondary: 76D05: Navier-Stokes equations [See also 35Q30] 60F99: None of the above, but in this section 35K55: Nonlinear parabolic equations 35A35: Theoretical approximation to solutions {For numerical analysis, see 65Mxx, 65Nxx} 35A40

**Keywords**

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

#### Citation

Kotelenez, Peter. A Stochastic Navier-Stokes Equation for the Vorticity of a Two-Dimensional Fluid. Ann. Appl. Probab. 5 (1995), no. 4, 1126--1160. doi:10.1214/aoap/1177004609. https://projecteuclid.org/euclid.aoap/1177004609