Archive for Rational Mechanics and Analysis

, Volume 199, Issue 2, pp 527–561 | Cite as

Homogenization and Enhancement for the G—Equation

Article

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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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de BrestBrest CedexFrance
  2. 2.Department of MathematicsDuke UniversityDurhamUSA
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA

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