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: 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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