Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales

Simplified effective equations for the large scale front propagation of turbulent reaction-diffusion equations are developed here in the 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-Jacobi equations. The subtle issues regarding the validity of a Huygens 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 important 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 thermal-diffusive approximation without any ad hoc approximations.