Enhancement of the Traveling Front Speeds in Reaction-Diffusion Equations with Advection

TL;DR

This paper provides rigorous lower bounds on the speed of traveling fronts in reaction-diffusion equations with advection, showing linear speed-up for percolating flows and sublinear for cellular flows.

Contribution

It establishes new lower bounds on front speeds in reaction-diffusion equations with passive advection for different flow types.

Findings

Percolating flows cause linear speed-up proportional to flow amplitude.

Cellular flows lead to a slower, U^{1/5} growth in front speed.

Rigorous bounds are derived for both KPP and ignition nonlinearities.

Abstract

We establish rigorous lower bounds on the speed of traveling fronts and on the bulk burning rate in reaction-diffusion equation with passive advection. The non-linearity is assumed to be of either KPP or ignition type. We consider two main classes of flows. Percolating flows, which are characterized by the presence of long tubes of streamlines mixing hot and cold material, lead to strong speed-up of burning which is linear in the amplitude of the flow, $U$. On the other hand the cellular flows, which have closed streamlines, are shown to produce weaker increase in reaction. For such flows we get a lower bound which grows as $U^{1/5}$ for a large amplitude of the flow.