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We study the limiting behavior of solutions to appropriately rescaled versions of the Allen-Cahn equation, a model for phase transition in polycrystalline material. We rigourously establish the existence in the limit of a phase-antiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of Evans-Spruck and Chen-Giga-Goto after the onset of geometric singularities.

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Simplified effective equations for the large scale front propagation of turbulent reaction4iffusion equations are developed here in lhe simplest prototypical situation involving advection by turbulent velocity fields with two separated scales. A rigorous theory for large scale front propagation is developed, utilizing PDE techniques for viscosity solutions together with homogenization theory for Hamilton-Jacnbi equations. The subtle issues regarding the validity of a Hnygens principle for the effective large scale front propagation as well as elementary upper and lower bounds on the propagating front are developed in this paper. Simple-examples involving small scale periodic shear layers are also presented here; they indicate that these elementary upper and lower bounds on front propagation are sharp. One i m p o m t consequence of the theory developed in this paper is that the authors are able to write down and rigorously justify the appropriate renormalized effective large scale front equations for premixed turbulent combustion with two-scale incompressible velocity fields within the thermaldiffusive approximation without any ad hoc approximations.