### Abstract

This paper, a sequel to the author's previous papers [J. Math. Kyoto Univ. 27 (1987), no. 4, 597–619; MR0916761; in Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), 303–334, Academic Press, Boston, MA, 1987; MR0933829], is concerned with the vorticity equation which is equivalent to the two-dimensional nonstationary Navier-Stokes equations. The vorticity can be interpreted as the expectation of the McKean process associated with the vorticity equation. There is an n -particle system of stochastic differential equations which is expected to approximate the McKean process.

The author proves that an n -particle system actually approximates the McKean process as n→∞ in the sense of distributions of processes. As a limit, a weak solution of the vorticity equation is constructed. In other words the author gives a rigorous derivation of the vorticity equation from an n -particle system as a propagation of chaos. No smallness assumptions on the Reynolds number are imposed. Since the velocity is determined by the vorticity through a convolution with a singular kernel, the stochastic equations involved in this paper have singularities. One should emphasize that the results do not follow from a general theory. This paper is a nice application of both analytic and probabilistic results in the above-mentioned papers.

The author proves that an n -particle system actually approximates the McKean process as n→∞ in the sense of distributions of processes. As a limit, a weak solution of the vorticity equation is constructed. In other words the author gives a rigorous derivation of the vorticity equation from an n -particle system as a propagation of chaos. No smallness assumptions on the Reynolds number are imposed. Since the velocity is determined by the vorticity through a convolution with a singular kernel, the stochastic equations involved in this paper have singularities. One should emphasize that the results do not follow from a general theory. This paper is a nice application of both analytic and probabilistic results in the above-mentioned papers.

Original language | English |
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Title of host publication | The IMA Volumes in Mathematics and its Applications. |

Subtitle of host publication | Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986) |

Publisher | Springer New York |

Pages | 117–126 |

Number of pages | 10 |

Volume | 9 |

Publication status | Published - 1987 |

Externally published | Yes |

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## Cite this

Osada, H. (1987). Limit points of empirical distributions of vortices with small viscosity. In

*The IMA Volumes in Mathematics and its Applications. : Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986)*(Vol. 9, pp. 117–126). Springer New York.