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Applied Math Seminar
Reaction-diffusion fronts in a random flow
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Some reaction-diffusion equations admit traveling wave solutions; these are simplified models of a combustion reaction or spreading animal species. When advection is included in the model, the asymptotic behavior of the propagating front depends significantly on the structure of the flow field. I will describe the propagation of fronts that arise from scalar, reaction-advection-diffusion models with the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. For a class of random flows, we establish an extension of the well-known variational representation for the asymptotic front speed, a non-random constant. I will discuss some analytical bounds on the front speed that can be derived from this representation. The variational representation suggests a method of numerically approximating the front speed, and I will demonstrate the results of numerical experiments based on this approach. The analysis makes use of large deviations estimates for the related diffusion process in a random environment, and the variational principle is expressed in terms of the principal Lyapunov exponent of an auxiliary evolution problem. |