Speed-up of combustion fronts in shear flows

TL;DR

This paper analyzes how shear flows with large amplitude accelerate reaction-diffusion fronts in infinite cylinders, providing explicit asymptotic growth rates and solving an open problem in the field.

Contribution

It proves the asymptotic linear growth of front speeds with flow amplitude and characterizes the limiting speeds explicitly, advancing understanding of reaction-diffusion-advection systems.

Findings

Unique speeds are asymptotically linear in flow amplitude.

Explicit characterization of the asymptotic growth rate.

Convergence of traveling fronts to degenerate solutions under certain conditions.

Abstract

This paper is concerned with the analysis of speed-up of reaction-diffusion-advection traveling fronts in infinite cylinders with periodic boundary conditions. The advection is a shear flow with a large amplitude and the reaction is nonnegative, with either positive or zero ignition temperature. The unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity, solving an open problem from \cite{b}. The asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem, and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional H{\"{o}}rmander-type condition on the flow.