Homogenization and enhancement for the G-equation

TL;DR

This paper analyzes the homogenization of the G-equation, demonstrating how oscillatory advection effects average out, leading to an effective anisotropic model with enhanced front velocity and convergence to the Wulff shape.

Contribution

It proves homogenization of the G-equation with oscillatory advection, provides convergence rates, and shows velocity enhancement and shape convergence under certain conditions.

Findings

Solutions homogenize as oscillation size diminishes.

Averaging can enhance the front velocity.

Level sets converge to the Wulff shape at long times.

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.