THE REGULARIZING EFFECTS OF RESETTING IN A PARTICLE SYSTEM FOR THE BURGERS EQUATION

• 成果类型: Article
• 署名作者: Iyer, Gautam; Novikov, Alexei
• 署名单位: Carnegie Mellon University; Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP586
• 发表日期: 2011
• 页码: 1468-1501

摘要

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 Sigma(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 -> infinity we get convergence to the solution of the viscous Burgers equation on long time intervals.