A random space-time birth particle method for 2d vortex equations with external field

Fontbona, J.; Méléard, S.

doi:10.1090/S0025-5718-08-02097-8

A random space-time birth particle method for 2d vortex equations with external field
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by J. Fontbona and S. Méléard PDF

Math. Comp. 77 (2008), 1525-1558 Request permission

Abstract:

We consider incompressible 2d Navier-Stokes equations in the whole plane with external nonconservative forces fields. The initial data and external field are functions assumed to satisfy only slight integrability properties. We develop a probabilistic interpretation of these equations based on the associated vortex equation, in order to construct a numerical particle method to approximate the solutions. More precisely, we relate the vortex equation with additional term to a nonlinear process with random space-time birth, which provides a probabilistic description of the creation of vorticity. We then introduce interacting particle systems defined for a regularized interaction kernel, whose births are chosen randomly in time and space. By a coupling method, we show that these systems are approximations of the nonlinear process and obtain precise convergence estimates. From this result, we deduce a stochastic numerical particle method to obtain the vorticity and also to recover the velocity field. The results are either pathwise or of weak convergence, depending on the integrability of the data. We illustrate our results with simulations.

References

Matania Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal. 128 (1994), no. 4, 329–358. MR 1308857, DOI 10.1007/BF00387712
Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
M. Bossy, Vitesse de convergence d’algorithmes particulaires stochastiques et application à l’équation de Burgers, Ph.D. thesis, Université de Provence, 1995.
Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
Haïm Brezis, Remarks on the preceding paper by M. Ben-Artzi: “Global solutions of two-dimensional Navier-Stokes and Euler equations” [Arch. Rational Mech. Anal. 128 (1994), no. 4, 329–358; MR1308857 (96h:35148)], Arch. Rational Mech. Anal. 128 (1994), no. 4, 359–360. MR 1308858, DOI 10.1007/BF00387713
Barbara Busnello, A probabilistic approach to the two-dimensional Navier-Stokes equations, Ann. Probab. 27 (1999), no. 4, 1750–1780. MR 1742887, DOI 10.1214/aop/1022677547
Marco Cannone, Ondelettes, paraproduits et Navier-Stokes, Diderot Editeur, Paris, 1995 (French). With a preface by Yves Meyer. MR 1688096
Alexandre J. Chorin and Jerrold E. Marsden, A mathematical introduction to fluid mechanics, 3rd ed., Texts in Applied Mathematics, vol. 4, Springer-Verlag, New York, 1993. MR 1218879, DOI 10.1007/978-1-4612-0883-9
Georges-Henri Cottet and Petros D. Koumoutsakos, Vortex methods, Cambridge University Press, Cambridge, 2000. Theory and practice. MR 1755095, DOI 10.1017/CBO9780511526442
J. Fontbona, Nonlinear martingale problems involving singular integrals, J. Funct. Anal. 200 (2003), no. 1, 198–236. MR 1974095, DOI 10.1016/S0022-1236(02)00068-X
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
B. Jourdain, Diffusion processes associated with nonlinear evolution equations for signed measures, Methodol. Comput. Appl. Probab. 2 (2000), no. 1, 69–91. MR 1783154, DOI 10.1023/A:1010059302049
B. Jourdain and S. Méléard, Propagation of chaos and fluctuations for a moderate model with smooth initial data, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 6, 727–766 (English, with English and French summaries). MR 1653393, DOI 10.1016/S0246-0203(99)80002-8
Benjamin Jourdain and Sylvie Méléard, Probabilistic interpretation and particle method for vortex equations with Neumann’s boundary condition, Proc. Edinb. Math. Soc. (2) 47 (2004), no. 3, 597–624. MR 2097263, DOI 10.1017/S0013091503000142
H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, École d’été de probabilités de Saint-Flour, XII—1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 143–303. MR 876080, DOI 10.1007/BFb0099433
C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys. 84 (1982), no. 4, 483–503. MR 667756
Sylvie Méléard, A trajectorial proof of the vortex method for the two-dimensional Navier-Stokes equation, Ann. Appl. Probab. 10 (2000), no. 4, 1197–1211. MR 1810871, DOI 10.1214/aoap/1019487613
Sylvie Méléard, Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data, Probab. Theory Related Fields 121 (2001), no. 3, 367–388. MR 1867427, DOI 10.1007/s004400100154
P.-A. Raviart, An analysis of particle methods, Numerical methods in fluid dynamics (Como, 1983) Lecture Notes in Math., vol. 1127, Springer, Berlin, 1985, pp. 243–324. MR 802214, DOI 10.1007/BFb0074532
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Alain-Sol Sznitman, Topics in propagation of chaos, École d’Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 165–251. MR 1108185, DOI 10.1007/BFb0085169