Communications in Mathematical Sciences

Periodic homogenization of the inviscid G-equation for incompressible flows

Jack Xin and Yifeng Yu

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 2 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.cms/1288725272

Mathematical Reviews number (MathSciNet)
MR2744920

Zentralblatt MATH identifier
1372.76085

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

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

Citation

Xin, Jack; Yu, Yifeng. Periodic homogenization of the inviscid G-equation for incompressible flows. Commun. Math. Sci. 8 (2010), no. 4, 1067--1078. https://projecteuclid.org/euclid.cms/1288725272


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