  G-equation is a popular level set model in turbulent combustion, and becomes
an advective mean curvature type evolution equation when curvature of a moving
flame in a fluid flow is considered: $$ G_t + \left(1-d\,
\mathrm{Div}{\frac{DG}{|DG|}}\right)_+|DG|+V(x)\cdot DG=0. $$ Here $d>0$ is the
Markstein number and the positive part $()_+$ is imposed to avoid a
non-physical negative laminar flame speed. For simplicity of presentation, we
focus mainly on the case when $V:\mathbb{R}^2\to \mathbb{R}^2$ is the two
dimensional cellular flow with Hamiltonian $H = \sin x_1 \, \sin x_2$ and
amplitude $A$. Our main result is that for any unit vector $p\in \mathbb{R}^2$,
there exists a positive number $\overline H(p)$ such that if $G(x,0)=p\cdot x$,
then $$ \left|G(x,t)-p\cdot x+\overline H(p)t\right|\leq C \quad \text{in
$\mathbb{R}^2\times [0,\infty)$} $$ for a constant $C$ depending only on the
Markstein number $d$ and the cellular flow amplitude $A$. The number $\overline
H(p)$ corresponds to the effective burning velocity in the physics literature.
The non-coercivity encountered here is one of the major difficulties for
homogenization of the mean curvature-type equations. To overcome it, we
introduce a new approach that combines PDE methods with a dynamical analysis of
the Kohn-Serfaty deterministic game characterization of the curvature
G-equation utilizing the streamline structure of cellular flows. Extension to
general two-dimensional incompressible flows is also discussed. In three
dimensional incompressible flows, the existence of $\overline H(p)$ might fail
when the flow intensity exceeds a bifurcation value even for simple shear flows
[32].
