Long Time Dynamics for Combustion in Random Media

TL;DR

This paper proves that long-term behavior of combustion processes in complex random media converges to a deterministic PDE, extending previous results to more general multidimensional, non-isotropic cases with various dependence structures.

Contribution

It provides the first proof of effective PDE convergence for non-isotropic multidimensional reactions with finite and some infinite dependence ranges.

Findings

Existence of deterministic front speeds in all directions.

Convergence of long-time dynamics to a homogeneous Hamilton-Jacobi equation.

Extension of previous isotropic results to non-isotropic multidimensional settings.

Abstract

We study long time dynamics of combustive processes in random media, modeled by reaction-diffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large space-time scale dynamics of solutions to these equations is almost surely governed by a different effective PDE, which should be a homogeneous Hamilton-Jacobi equation. While this was previously proved in one dimension as well as for isotropic reactions in several dimensions (i.e., with radially symmetric laws), we provide here the first proof of this phenomenon in the general non-isotropic multidimensional setting. Our results hold for reactions that have finite ranges of dependence (i.e., their values are independent at sufficiently distant points in space) as well as for some with infinite ranges of dependence, and are based on proving existence of deterministic front (propagation) speeds in all directions for these reactions.