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July 2011 The regularizing effects of resetting in a particle system for the Burgers equation
Gautam Iyer, Alexei Novikov
Ann. Probab. 39(4): 1468-1501 (July 2011). DOI: 10.1214/10-AOP586

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.

Citation

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Gautam Iyer. Alexei Novikov. "The regularizing effects of resetting in a particle system for the Burgers equation." Ann. Probab. 39 (4) 1468 - 1501, July 2011. https://doi.org/10.1214/10-AOP586

Information

Published: July 2011
First available in Project Euclid: 5 August 2011

zbMATH: 1248.60071
MathSciNet: MR2857247
Digital Object Identifier: 10.1214/10-AOP586

Subjects:
Primary: 60H15
Secondary: 35L67 , 65C35

Keywords: Burgers’ equations , stochastic Lagrangian

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 4 • July 2011
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