Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds

TL;DR

This paper proves the existence of KPP traveling fronts in space-time periodic incompressible flows and derives a variational formula for their minimal speeds using a novel dynamic approach that avoids degeneracy issues.

Contribution

It introduces a dynamic method to establish the existence of KPP fronts and derives a variational formula for minimal speeds in complex advection fields.

Findings

Existence of KPP fronts in space-time periodic incompressible flows.

A variational formula for minimal propagation speeds.

A dynamic approach that bypasses degeneracy in the equations.

Abstract

We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We apply a dynamic argument to justify that the constructed KPP traveling fronts propagate at minimal speeds, and derive the speed variational formula. The dynamic method avoids the degeneracy in traveling front equations, and utilizes the parabolic maximum principle of the governing reaction-diffusion-advection equation. The dynamic method does not rely on existence of traveling fronts.