The Annals of Probability

The regularizing effects of resetting in a particle system for the Burgers equation

Gautam Iyer and Alexei Novikov

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Abstract

We study the dissipation mechanism of a stochastic particle system for the Burgers equation. The velocity field of the viscous Burgers and Navier–Stokes equations can be expressed as an expected value of a stochastic process based on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3 (2008) 330–345]. In this paper we study a particle system for the viscous Burgers equations using a Monte–Carlo version of the above; we consider N copies of the above stochastic flow, each driven by independent Wiener processes, and replace the expected value with 1/N times the sum over these copies. A similar construction for the Navier–Stokes equations was studied by Mattingly and the first author of this paper [Iyer and Mattingly Nonlinearity 21 (2008) 2537–2553].

Surprisingly, for any finite N, the particle system for the Burgers equations shocks almost surely in finite time. In contrast to the full expected value, the empirical mean 1/N ∑1N does not regularize the system enough to ensure a time global solution. To avoid these shocks, we consider a resetting procedure, which at first sight should have no regularizing effect at all. However, we prove that this procedure prevents the formation of shocks for any N ≥ 2, and consequently as N → ∞ we get convergence to the solution of the viscous Burgers equation on long time intervals.

Article information

Source
Ann. Probab. Volume 39, Number 4 (2011), 1468-1501.

Dates
First available in Project Euclid: 5 August 2011

Permanent link to this document
http://projecteuclid.org/euclid.aop/1312555805

Digital Object Identifier
doi:10.1214/10-AOP586

Zentralblatt MATH identifier
05955099

Mathematical Reviews number (MathSciNet)
MR2857247

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 65C35: Stochastic particle methods [See also 82C80] 35L67: Shocks and singularities [See also 58Kxx, 76L05]

Keywords
Burgers’ equations stochastic Lagrangian

Citation

Iyer, Gautam; Novikov, Alexei. The regularizing effects of resetting in a particle system for the Burgers equation. The Annals of Probability 39 (2011), no. 4, 1468--1501. doi:10.1214/10-AOP586. http://projecteuclid.org/euclid.aop/1312555805.


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References

  • [1] Alibaud, N., Droniou, J. and Vovelle, J. (2007). Occurrence and non-appearance of shocks in fractal Burgers equations. J. Hyperbolic Differ. Equ. 4 479–499.
  • [2] Beale, J. T., Kato, T. and Majda, A. (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94 61–66.
  • [3] Constantin, P. and Iyer, G. (2008). A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations. Comm. Pure Appl. Math. 61 330–345.
  • [4] De Lellis, C., Otto, F. and Westdickenberg, M. (2004). Minimal entropy conditions for Burgers equation. Quart. Appl. Math. 62 687–700.
  • [5] Evans, L. C. (1998). Partial Differential Equations. Graduate Studies in Mathematics 19. Amer. Math. Soc., Providence, RI.
  • [6] Fathi, A. (1997). Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math. 324 1043–1046.
  • [7] Iyer, G. (2006). A stochastic perturbation of inviscid flows. Comm. Math. Phys. 266 631–645.
  • [8] Iyer, G. (2006). A stochastic Lagrangian formulation of the Navier–Stokes and related transport equations. Ph.D. thesis, Univ. Chicago.
  • [9] Iyer, G. and Mattingly, J. (2008). A stochastic-Lagrangian particle system for the Navier–Stokes equations. Nonlinearity 21 2537–2553.
  • [10] Jauslin, H. R., Kreiss, H. O. and Moser, J. (1999). On the forced Burgers equation with periodic boundary conditions. In Differential Equations: La Pietra 1996 (Florence). Proc. Sympos. Pure Math. 65 133–153. Amer. Math. Soc., Providence, RI.
  • [11] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [12] Kiselev, A., Nazarov, F. and Shterenberg, R. (2008). Blow up and regularity for fractal Burgers equation. Dyn. Partial Differ. Equ. 5 211–240.
  • [13] Krylov, N. V. (1999). An analytic approach to SPDEs. In Stochastic Partial Differential Equations: Six Perspectives. Math. Surveys Monogr. 64 185–242. Amer. Math. Soc., Providence, RI.
  • [14] Krylov, N. V. and Rozovskiĭ, B. L. (1982). Stochastic partial differential equations and diffusion processes. Uspekhi Mat. Nauk 37 75–95.
  • [15] Krylov, N. V. (2007). Maximum principle for SPDEs and its applications. In Stochastic Differential Equations: Theory and Applications. Interdiscip. Math. Sci. 2 311–338. World Sci. Publ., Hackensack, NJ.
  • [16] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [17] Majda, A. J. and Bertozzi, A. L. (2002). Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics 27. Cambridge Univ. Press, Cambridge.
  • [18] Metropolis, N. and Ulam, S. (1949). The Monte Carlo method. J. Amer. Statist. Assoc. 44 335–341.
  • [19] Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
  • [20] Rozovskiĭ, B. L. (1990). Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Translated from the Russian by A. Yarkho. Mathematics and Its Applications (Soviet Series) 35. Kluwer, Dordrecht.
  • [21] Weinan, E., Khanin, K., Mazel, A. and Sinai, Y. (2000). Invariant measures for Burgers equation with stochastic forcing. Ann. of Math. (2) 151 877–960.
  • [22] Yudovich, V. I. (1963). Non-stationary flows of an ideal incompressible fluid. Z. Vychisl. Mat. i Mat. Fiz. 3 1032–1066.