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Hydrodynamics in two dimensions and vortex theory

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Abstract

We consider the Navier-Stokes equation for a viscous and incompressible fluid inR 2. We show that such an equation may be interpreted as a mean field equation (Vlasov-like limit) for a system of particles, called vortices, interacting via a logarithmic potential, on which, in addition, a stochastic perturbation is acting. More precisely we prove that the solutions of the Navier-Stokes equation may be approximated, in a suitable way, by finite dimensional diffusion processes with the diffusion constant related to the viscosity. As a particular case, when the diffusion constant is zero, the finite dimensional theory reduces to the usual deterministic vortex theory, and the limiting equation reduces to the Euler equation.

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Authors and Affiliations

  1. Dipartimento di Matematica, Libera Università di Trento, I-38050, Povo (Trento), Italy

    C. Marchioro

  2. Istituto Matematico dell'Università di Roma, I-00100, Roma, Italy

    M. Pulvirenti

Authors
  1. C. Marchioro
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  2. M. Pulvirenti
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Communicated by J. Glimm

Partially supported by Italian CNR

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Marchioro, C., Pulvirenti, M. Hydrodynamics in two dimensions and vortex theory. Commun.Math. Phys. 84, 483–503 (1982). https://doi.org/10.1007/BF01209630

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  • DOI: https://doi.org/10.1007/BF01209630

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