Limit points of empirical distributions of vortices with small viscosity

研究成果: 著書/レポートタイプへの貢献会議での発言

抄録

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.
元の言語英語
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)
出版者Springer New York
ページ117–126
ページ数10
9
出版物ステータス出版済み - 1987
外部発表Yes

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

これを引用

Osada, H. (1987). 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) (巻 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). 巻 9 Springer New York, 1987. p. 117–126.

研究成果: 著書/レポートタイプへの貢献会議での発言

Osada, H 1987, 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). 巻. 9, Springer New York, pp. 117–126.
Osada H. 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). 巻 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). 巻 9 Springer New York, 1987. pp. 117–126
@inproceedings{15a53250ea7145e8b01554e1d9da62bf,
title = "Limit points of empirical distributions of vortices with small viscosity",
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.",
author = "Hirofumi Osada",
year = "1987",
language = "English",
volume = "9",
pages = "117–126",
booktitle = "The IMA Volumes in Mathematics and its Applications.",
publisher = "Springer New York",
address = "United States",

}

TY - GEN

T1 - Limit points of empirical distributions of vortices with small viscosity

AU - Osada, Hirofumi

PY - 1987

Y1 - 1987

N2 - 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.

AB - 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.

M3 - Conference contribution

VL - 9

SP - 117

EP - 126

BT - The IMA Volumes in Mathematics and its Applications.

PB - Springer New York

ER -