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Convergence of the random vortex method. (English) Zbl 0635.35077

An approximate solution of the Cauchy problem for the incompressible Navier-Stokes equations in two space dimensions is constructed by means of a modified random vortex method. The main result is the following convergence theorem: if the initial vorticity distribution \(\omega\) (x,0)\(\in S\) (the Schwartz class), then with high probability this method produces good approximations to the true velocity.
Reviewer: J.R.Romanovsky

MSC:

35Q30 Navier-Stokes equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65 Brownian motion
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Anderson, SIAM J. Num. Anal. 22 pp 413– (1985)
[2] Stochastic Differential Equations: Theory and Applications, Wiley, New York. 1974.
[3] and , Rates of convergence for viscous splitting of the Navier-Stokes equations, Math. Comp., 1981, pp. 243–259. · Zbl 0518.76027
[4] Beak, Math. Comp. 39 pp 1– (1982)
[5] Beale, Math. Comp. 39 pp 29– (1982)
[6] and , A Mathematical Introduction to Fluid Mechanics, Springer Verlag, New York, 1979. · Zbl 0417.76002
[7] A Course in Probability Theory, Academic Press, New York, 1974.
[8] PhD Thesis, l’Université Pierre et Marie Curie, 1982.
[9] Introduction to Partial Differential Equations, Princeton University Press, Princeton, New Jersey, 1976. · Zbl 0325.35001
[10] Hald, SIAM J. Num. Anal. 16 pp 726– (1979)
[11] Hald, SIAM J. Sci. Stat. Comp. 2 pp 85– (1981)
[12] Hörmander, Acta Math. 127 pp 79– (1971)
[13] Marchioro, Comm. Math. Phys. 84 pp 483– (1982)
[14] Propagation of chaos for a class of nonlinear parabolic equations, in Lecture Series in Differential Equations, Session 7, Catholic University Press, 1967.
[15] Moser, Ann. Sco. Norm. Pisa 20 pp 265– (1966)
[16] Propagation of chaos for the two-dimensional Navier-Stokes equation, preprint.
[17] Rosenhead, Proc. Roy. Soc. Lond. A. 134 pp 170– (1931)
[18] personal communication.
[19] Singular Integrals and Differentiability Properties of Functions. Princeton University Press. Princeton, New Jersey, 1970. · Zbl 0207.13501
[20] and , Multidimensional Diffusion Processes, Springer, New York, 1979. · Zbl 0426.60069
[21] A propagation of chaos result for Burgers’ equation, preprint.
[22] Pseudodifferential Operators, Princeton University Press, Princeton, New Jersey, 1981. · Zbl 0453.47026
[23] Introduction to Pseudodifferential and Fourier Integral Operators, Vol. I, Plenum Press, New York, 1980. · Zbl 0453.47027
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