KPP Fronts in Random Media

Abstract

In this chapter, we discuss KPP fronts in space and/or time random flows by a combination of PDE and probabilistic methods. We first consider fronts in spatial random shear flows in a channel domain with finite width L. Here randomness appears in the transverse direction of the front. The variational principle (2.52) under zero Neumann boundary condition applies to each random speed, and allows a combined PDE-probabilistic analysis and a computational study of the speed ensemble. We identify new random phenomena such as the resonance in speed dependence on correlation length of the flows, speed slowdown due to temporal decorrelation of random flows, and speed divergence due to extreme behavior of random media. For example, for channel domain width L1, front speed diverges due to unbounded running maxima of the shear flow, which shares the same source of divergence as stochastic homogenization of Hamilton-Jacobi equations in Chapter 4. We shall present front growth or decay laws in random media, in comparison with those of periodic media in Chapter 3. Finally, we outline a recent breakthrough in solving the turbulent front speed problem for the KPP model, and study related speed bounds.