Front Quenching in the G-equation Model Induced by Straining of Cellular Flow

We study homogenization of the G-equation with a flow straining term (or the strain G-equation) in two dimensional periodic cellular flow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton–Jacobi equation. The main objective is to investigate how the flow induced straining (the nonconvex term) influences front propagation as the flow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent flame speed s _T (effective Hamiltonian) is well-defined for any periodic flow with small divergence and is enhanced by the cellular flow as s _T ≧ O(A/log A). In the second regime where A is slightly above the critical value, homogenization breaks down, and s _T is not well-defined along any direction. Solutions become a mixture of a fast moving part and a stagnant part. When A is sufficiently large, the whole flame front ceases to propagate forward due to the flow induced straining. In particular, along directions p = (±1, 0) and (0, ±1), s _T is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player differential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.