A Variational Principle Based Study of Random Front Speeds

University Of California-Irvine, Irvine CA

Investigators

Abstract

Abstract: DMS-0506766, J Xin, University of Texas Title: A Variational Principle Based Study of Random Front Speeds The project pursues analysis and computation of reaction-diffusion front speeds in random media based on their variational principles. Direct computation of the speed ensemble can be both expensive and less accurate. Because front speeds are part of the large time (large scale) behavior of solutions to stochastic reaction-diffusion equations, one needs a large domain size, sufficient resolution of front structures, and many realizations of random samples. The variational principles are established by exploiting the analytical and probabilistic properties of solutions. Significant dimensional reductions are achieved so that only associated linear problems need to be solved to find principal eigenvalues or Lyapunov exponents. Besides being a useful tool for analysis of speed statistics, the variational principles also help to generate fast and efficient computational algorithms. The project will explore this approach to study front speeds in various space/time random media, the speed asymptotic laws and the dependence on statistics of random media. The project is motivated by flame fronts in the environment (forest or building fires) and internal combustion engines of vehicles, where fluid (air or liquid) motion could alter the speed of burning process significantly. The fluid motion often contains uncertainties and can be best described as random media. Scientific understanding and efficient computing of front speeds can help to control the spread of flames, and minimize the waste gases from the combustion engines to benefit the environment. The methods being developed in the project will contribute to both the understanding and computing of random fronts, a subject largely in its infancy.