Quantitative Homogenization for Combustion in Random Media

TL;DR

This paper establishes the first quantitative stochastic homogenization results for reaction-diffusion equations with ignition reactions in low dimensions, demonstrating algebraic convergence rates to homogenized limits characterized by Hamilton-Jacobi equations.

Contribution

It provides the first quantitative homogenization results for reaction-diffusion equations with ignition reactions in dimensions up to three, including convergence rates.

Findings

Algebraic rate of convergence to homogenized solutions.

Homogenized limits are viscosity solutions of Hamilton-Jacobi equations.

Results apply to reactions with finite dependence ranges or close to such reactions.

Abstract

We obtain the first quantitative stochastic homogenization result for reaction-diffusion equations, for ignition reactions in dimensions $d\le 3$ that either have finite ranges of dependence or are close enough to such reactions, and for solutions with initial data that approximate characteristic functions of general convex sets. We show algebraic rate of convergence of these solutions to their homogenized limits, which are (discontinuous) viscosity solutions of certain related Hamilton-Jacobi equations.