A Variational Principle Based Study of Random Front Speeds

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Abstract: DMS-0506766, J Xin, University of TexasTitle: A Variational Principle Based Study of Random Front Speeds The project pursues analysis and computation of reaction-diffusionfront speeds in random media based on their variational principles.Direct computation of the speed ensemble can be both expensiveand 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 establishedby exploiting the analytical and probabilistic properties of solutions. Significant dimensional reductions are achieved so that onlyassociated linear problems need to be solved to find principaleigenvalues 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 willexplore 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 processsignificantly. The fluid motion often contains uncertainties andcan be best described as random media. Scientific understandingand efficient computing of front speeds can help to control thespread of flames, and minimize the waste gases from the combustion enginesto benefit the environment. The methods being developed in the project will contribute to both the understanding and computingof random fronts, a subject largely in its infancy.