Abstract
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.
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Communicated by P.-L. Lions
P. E. Souganidis was partially supported by the National Science Foundation. P. Cardaliaguet was partially supported by the ANR (Agence Nationale de la Recherche) through the MICA project (ANR-06-BLAN-0082).
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Cardaliaguet, P., Nolen, J. & Souganidis, P.E. Homogenization and Enhancement for the G—Equation. Arch Rational Mech Anal 199, 527–561 (2011). https://doi.org/10.1007/s00205-010-0332-8
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DOI: https://doi.org/10.1007/s00205-010-0332-8