Jack Xin: G-equations and Front Motion in Fluid Flows

G-equations are level set Hamilton-Jacobi equations (HJE) for modeling flame fronts in turbulent combustion where a fundamental problem is to characterize the turbulent flame speeds s_T. The existence of s_T is connected with the homogenization of HJE, however classical theory does not apply due to the non-coercive and non-convex nature of the level set Hamiltonian. We shall illustrate the asymptotic properties of s_T from both Eulerian and Lagrangian perspectives in the case of two dimensional periodic incompressible flows, in particular cellular flows. Analytical and numerical results demonstrate that G-equations capture well the enhancement, slow down and quenching phenomena observed in fluid experiments. We also comment on s_T in chaotic flows. This is joint work with Yifeng Yu and Yu-Yu Liu.