Limit points of empirical distributions of vortices with small viscosity

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.
Original languageEnglish
Title of host publicationThe IMA Volumes in Mathematics and its Applications.
Subtitle of host publicationHydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986)
PublisherSpringer New York
Pages117–126
Number of pages10
Volume9
Publication statusPublished - 1987
Externally publishedYes

Fingerprint

Empirical Distribution
Limit Point
Vorticity
Vortex
Viscosity
Particle System
Stochastic Equations
Propagation of Chaos
Singular Kernel
Probabilistic Methods
Weak Solution
Reynolds number
Convolution
Navier-Stokes Equations
Physics
Singularity
Differential equation

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.

Limit points of empirical distributions of vortices with small viscosity. / Osada, Hirofumi.

The IMA Volumes in Mathematics and its Applications. : Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986). Vol. 9 Springer New York, 1987. p. 117–126.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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, Springer New York, pp. 117–126.
Osada H. 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. Springer New York. 1987. p. 117–126
Osada, Hirofumi. / Limit points of empirical distributions of vortices with small viscosity. The IMA Volumes in Mathematics and its Applications. : Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986). Vol. 9 Springer New York, 1987. pp. 117–126
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