Abstract: G-equation is a well known level set model in turbulent combustion, and becomes an advective mean curvature type evolution equation when the curvature effect is considered:
$$
G_t + \left(1-d\, \Div{\frac{DG}{|DG|}}\right)_+|DG|+V(x)\cdot DG=0.
$$
 In this talk, I will show the existence of effective burning velocity under the above curvature G-equation model when $V$ is a two dimensional cellular flow, which can be extended to more general two dimensional incompressible periodic flows.  Our proof combines PDE methods with a dynamical analysis of the Kohn-Serfaty deterministic game characterization of the curvature G-equation based on the two dimensional structures.  In three dimensions,  the effective burning velocity will cease to exist even for simple periodic shear flows when the flow intensity surpasses a bifurcation value. The existence result is based on joint work with  Hongwei Gao, Ziang Long and Jack Xin, while the non-existence result is  from collaboration with Hiro Mitake,  Connor Mooney, Hung Tran, and Jack Xin.