# Pulsating Front Speed-up and Quenching of Reaction by Fast Advection

## TL;DR

This paper investigates how fast periodic incompressible flows with cellular structures can accelerate reaction fronts in reaction-diffusion equations, revealing that the flow geometry determines speed-up and quenching capabilities.

## Contribution

It establishes that front speed-up depends solely on flow geometry and not on the reaction type, and characterizes flows capable of quenching ignition reactions.

## Key findings

- Front speed-up occurs if and only if flow geometry allows it.
- Speed-up is independent of the specific reaction function.
- Flows capable of quenching ignition reactions are precisely those that speed up fronts.

## Abstract

We consider reaction-diffusion equations with combustion-type non-linearities in two dimensions and study speed-up of their pulsating fronts by general periodic incompressible flows with a cellular structure. We show that the occurence of front speed-up in the sense $\lim_{A\to\infty} c_*(A)=\infty$, with $A$ the amplitude of the flow and $c_*(A)$ the (minimal) front speed, only depends on the geometry of the flow and not on the reaction function. In particular, front speed-up happens for KPP reactions if and only if it does for ignition reactions. We also show that the flows which achieve this speed-up are precisely those which, when scaled properly, are able to quench any ignition reaction.