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Periodic homogenization of the inviscid G-equation for incompressible flows. (English) Zbl 1372.76085
Summary: 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 [Y.-Y. Li and the authors, Nonlinearity 23, No. 10, 2351–2367 (2010; Zbl 1197.70012)]. 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.

76M50 Homogenization applied to problems in fluid mechanics
35F21 Hamilton-Jacobi equations
76V05 Reaction effects in flows
70H20 Hamilton-Jacobi equations in mechanics
Full Text: DOI Euclid