Limit points of empirical distributions of vortices with small viscosity

研究成果: Chapter in Book/Report/Conference proceedingConference contribution


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.
ホスト出版物のタイトルThe IMA Volumes in Mathematics and its Applications.
ホスト出版物のサブタイトルHydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986)
出版社Springer New York
出版ステータス出版済み - 1987


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