Homogenization and Enhancement for the G—Equation
Nolen, James; Cardaliaguet, Pierre; Souganidis, Panagiotis E. (2011), Homogenization and Enhancement for the G—Equation, Archive for Rational Mechanics and Analysis, 199, 2, p. 527-561. http://dx.doi.org/10.1007/s00205-010-0332-8
TypeArticle accepté pour publication ou publié
Journal nameArchive for Rational Mechanics and Analysis
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Abstract (EN)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.
Subjects / KeywordsG-equation; Hamilton-Jacobi equation
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