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Homogenization and enhancement for the \(G\)-equation. (English) Zbl 1294.35002
Summary: 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.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35F21 Hamilton-Jacobi equations
35Q35 PDEs in connection with fluid mechanics
49J45 Methods involving semicontinuity and convergence; relaxation
76M50 Homogenization applied to problems in fluid mechanics
76V05 Reaction effects in flows
80A25 Combustion
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