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Publication 09-CNA-21

The Regularizing Effects of Resetting in a Particle System for the Burgers' Equation

Gautam Iyer
Carnegie Mellon
Department of Mathematical Sciences
Pittsburgh, PA 15213
gautam@math.cmu.edu

Alexei Novikov
Penn State University
Department of Mathematics
University Park, PA 16803
anovikov@math.psu.edu

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, Iyer, Comm. Pure Appl. Math, 2008). 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 $\frac{1}{N}$ times the sum over these copies. A similar construction for the Navier-Stokes equations was studied by J. Mattingly and the first author (Nonlinearity, 2008).

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 $\frac{1}{ N}\ \sum^N_1$ 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 $N \geq 2$, and consequently as $N \rightarrow \infty$ we get convergence to the solution of the viscous Burgers. equation on long time intervals.

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