Communications in Mathematical Sciences

Periodic homogenization of the inviscid G-equation for incompressible flows

Jack Xin and Yifeng Yu

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G-equations are popular front propagation models in combustion literature and describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends a two space dimensional result in "Periodic homogenization of G-equations and viscosity effects," Nonlinearity, to appear. We construct approximate correctors to bypass the lack of compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on a local reachability property of the controlled flow trajectory as well as incompressibility of the flow. Homogenization then follows from the comparison principle and the perturbed test function method. The effective Hamiltonian is convex and homogeneous of degree one. It is also coercive if we further assume that the flow is mean zero.

Article information

Commun. Math. Sci., Volume 8, Number 4 (2010), 1067-1078.

First available in Project Euclid: 2 November 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 70H20: Hamilton-Jacobi equations 76M50: Homogenization 76M45: Asymptotic methods, singular perturbations 76N20: Boundary-layer theory

Non-coercive Hamilton-Jacobi equations controlled flow local reachability incompressibility periodic homogenization


Xin, Jack; Yu, Yifeng. Periodic homogenization of the inviscid G-equation for incompressible flows. Commun. Math. Sci. 8 (2010), no. 4, 1067--1078.

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