# Limit Points of Empirical Distributions of Vorticies with Small Viscosity

Chapter
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 9)

## Abstract

Let v(t,z) (z=(x,y)εR2) be the vorticity of an incompressible and viscous two dimensional fluid, under the action of an external conservative field. Then v is described by the following evolution equation
$${{\partial }_{\text{t}}}\text{v+}\left( \text{u}\centerdot \Delta \right)\text{v}-v\Delta \text{v=0, u}\left( \text{t},\text{z} \right)=\left( \Delta {}^{\bot }\text{G} \right)*\text{v}\left( \text{t},\text{z} \right),$$
(0.1)
where $$\text{G}\left( \text{z} \right)=-{{\left( 2\text{ }\!\!\pi\!\!\text{ } \right)}^{-1}}\text{ log}\left| \text{z} \right|,*$$ denotes convolution, $$\Delta =\left( \frac{\partial }{\partial \text{x}},\frac{\partial }{\partial \text{y}} \right)$$ and $${{\Delta }^{\bot }}=\left( \frac{\partial }{\partial \text{y}},-\frac{\partial }{\partial \text{x}} \right)$$. Here v > 0 denotes the viscosity constant. As far as strong solutions concerns, (0,1) is equivalent to the Navier-Stokes equation. In fact, u(t,z) turns to be the velocity field described by the Navier-Stokes equation. Conversely we can get v from u as v = curl u. Since the two dimensional Navier-Stokes equation is an equation of a vector valued function, a probabilistic treatment is not easy, while the vorticity equation (0.1) is nothing but a McKean’s type non-linear equation (see [3]. Such an observation for the two dimensional Navier-Stokes equation was made by Marchioro-Pulvirenti in [2].

## Keywords

Vorticity Equation Vortex Method Small Viscosity Homogenous Boltzmann Equation Dimensional Fluid

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