  We consider the so-called G-equation, a level set Hamilton-Jacobi equation,
used as a sharp interface model for flame propagation, perturbed by an
oscillatory advection in a spatio-temporal periodic environment. Assuming that
the advection has suitably small spatial divergence, we prove that, as the size
of the oscillations diminishes, the solutions homogenize (average out) and
converge to the solution of an effective anisotropic first-order
(spatio-temporal homogeneous) level set equation. Moreover we obtain a rate of
convergence and show that, under certain conditions, the averaging enhances the
velocity of the underlying front. We also prove that, at scale one, the level
sets of the solutions of the oscillatory problem converge, at long times, to
the Wulff shape associated with the effective Hamiltonian. Finally we also
consider advection depending on position at the integral scale.
