Asymptotics for Turbulent Flame Speeds of the Viscous G-Equation Enhanced by Cellular and Shear Flows

G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton-Jacobi equations with convex (L¹ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue H̄ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed s_T. An important problem in turbulent combustion theory is to study properties of s_T, in particular how s_T depends on the flow amplitude A. In this paper, we study the behavior of as A → + ∞ at any fixed diffusion constant d > 0. For cellular flow, we show that H̄ (A,d) ≦ C(d) for all d > 0,where C(d) is a constant depending on d, but independent of A. Compared with H̄(A,0) = O(A/log A), A ≫ 1, of the inviscid G-equation (d = 0), presence of diffusion dramatically slows down front propagation. For shear flow, where λ (d) is strictly decreasing in d, and has zero derivative at d = 0. The linear growth law is also valid for s_T of the curvature dependent G-equation in shear flows.