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Limit Points of Empirical Distributions of Vorticies with Small Viscosity

  • Hirofumi Osada
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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Hirofumi Osada
    • 1
  1. 1.Department of MathematicsNara Women’s UniversityKita-Uoya Nishimachi, Nara 630Japan

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