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 sT. An important problem in turbulent combustion theory is to study properties of sT, in particular how sT 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 sT of the curvature dependent G-equation in shear flows.
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