  It is well known in the combustion community that curvature effect in general
slows down flame propagation speeds because it smooths out wrinkled flames.
However, such a folklore has never been justified rigorously. In this paper, as
the first theoretical result in this direction, we prove that the turbulent
flame speed (an effective burning velocity) is decreasing with respect to the
curvature diffusivity (Markstein number) for shear flows in the well known
G-equation model. Our proof involves several novel and rather sophisticated
inequalities arising from the nonlinear structure of the equation. On a related
fundamental issue, we solve the selection problem of weak solutions or find the
"physical fluctuations" when the Markstein number goes to zero and solutions
approach those of the inviscid G-equation model. The limiting solution is given
by a closed form analytical formula.
