### 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 ∑^{N}_{1} 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.

Original language | English (US) |
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Pages (from-to) | 1468-1501 |

Number of pages | 34 |

Journal | Annals of Probability |

Volume | 39 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2011 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Probability*,

*39*(4), 1468-1501. https://doi.org/10.1214/10-AOP586