Source: Ann. Probab.
Volume 39, Number 4
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.
 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.
 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.
Mathematical Reviews (MathSciNet): MR763762
 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.
 De Lellis, C., Otto, F. and Westdickenberg, M. (2004). Minimal entropy conditions for Burgers equation. Quart. Appl. Math. 62 687–700.
 Evans, L. C. (1998). Partial Differential Equations. Graduate Studies in Mathematics 19. Amer. Math. Soc., Providence, RI.
 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.
 Iyer, G. (2006). A stochastic perturbation of inviscid flows. Comm. Math. Phys. 266 631–645.
 Iyer, G. (2006). A stochastic Lagrangian formulation of the Navier–Stokes and related transport equations. Ph.D. thesis, Univ. Chicago.
 Iyer, G. and Mattingly, J. (2008). A stochastic-Lagrangian particle system for the Navier–Stokes equations. Nonlinearity 21 2537–2553.
 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.
 Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
 Kiselev, A., Nazarov, F. and Shterenberg, R. (2008). Blow up and regularity for fractal Burgers equation. Dyn. Partial Differ. Equ. 5 211–240.
 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.
 Krylov, N. V. and Rozovskiĭ, B. L. (1982). Stochastic partial differential equations and diffusion processes. Uspekhi Mat. Nauk 37 75–95.
Mathematical Reviews (MathSciNet): MR683274
 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.
 Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
 Majda, A. J. and Bertozzi, A. L. (2002). Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics 27. Cambridge Univ. Press, Cambridge.
 Metropolis, N. and Ulam, S. (1949). The Monte Carlo method. J. Amer. Statist. Assoc. 44 335–341.
Mathematical Reviews (MathSciNet): MR31341
 Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
 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.
 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.
 Yudovich, V. I. (1963). Non-stationary flows of an ideal incompressible fluid. Z. Vychisl. Mat. i Mat. Fiz. 3 1032–1066.
Mathematical Reviews (MathSciNet): MR158189