KPP Front Speeds in Random Shears and the Parabolic Anderson Problem

TL;DR

This paper investigates how random shear flows influence the speed of reaction fronts in KPP equations, revealing quadratic enhancement for small amplitudes and divergence for time-independent fields, through connections with the parabolic Anderson problem.

Contribution

It establishes a novel link between KPP front speeds, Lyapunov exponents, and homogenized Hamiltonians, providing new asymptotic formulas for different shear regimes.

Findings

Quadratic growth of front speeds with small rms shear amplitudes.

Logarithmic divergence of front speeds for time-independent shears.

Connection of front speed asymptotics with the parabolic Anderson problem.

Abstract

We study the asymptotics of front speeds of the reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity and zero mean stationary ergodic Gaussian shear advection on the entire plane. By exploiting connections of KPP front speeds with the almost sure Lyapunov exponents of the parabolic Anderson problem, and with the homogenized Hamiltonians of Hamilton-Jacobi equations, we show that front speeds enhancement is quadratic in the small root mean square (rms) amplitudes of white in time zero mean Gaussian shears, and it grows at the order of the large rms amplitudes. However, front speeds diverge logarithmically if the shears are time independent zero mean stationary ergodic Gaussian fields.