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


35Q30 Navier-Stokes equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65 Brownian motion
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.