zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[1] Alvarez O., Ishii H.: Hamilton-Jacobi equations with partial gradient and application to homogenization. Comm. Partial Differential Equations 26(5–6), 983–1002 (2001) · Zbl 1014.49021
[2] Alvarez O., Cardaliaguet P., Monneau R.: Existence and uniqueness for dislocation dynamics with nonnegative velocity. Interfaces Free Bound. 7(4), 415–434 (2005) · Zbl 1099.35148
[3] Alvarez O., Bardi M.: Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, to appear in Memoirs of the Amer. Math. Soc · Zbl 1209.35001
[4] Arisawa M., Lions P.-L.: On ergodic stochastic control. Comm. Partial Differential Equations 23(11–12), 2187–2217 (1998) · Zbl 1126.93434
[5] Barles G.: Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications (Berlin), 17, Springer-Verlag, Paris, 1994
[6] Barles G., Soner H.M., Souganidis, P.E.: Front propagation and phase field theory. SIAM J. Control Optim. 31(2), 439–469 (1993) · Zbl 0785.35049
[7] Barles G.: Some homogenization results for non-coercive Hamilton-Jacobi equations. Calculus of Variations and Partial Differential Equations 30(4), 449–466 (2007) · Zbl 1136.35004
[8] Barron E.N., Jensen R.: Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial Differential Equations 15(12), 1713–1742 (1990) · Zbl 0732.35014
[9] Cannarsa P., Frankowska H.: Interior sphere property of attainable sets and time optimal control problems. ESAIM Control Optim. Calc. Var. 12(2), 350–370 (2006) (electronic) · Zbl 1105.93007
[10] Capuzzo-Dolcetta I., Ishii H.: On the rate of convergence in homogenization of Hamilton-Jacobi equations. Indiana Univ. Math. J. 50(3), 1113–1129 (2001) · Zbl 1256.35003
[11] Cardaliaguet P.: Ergodicity of Hamilton-Jacobi equations with a non-coercive non-convex Hamiltonian in $${\(\backslash\)mathbb R\^2/\(\backslash\)mathbb Z\^2}$$ . To appear in Ann. Inst. H. Poincaré, Anal. Non Linéaire
[12] Cencini M., Torcini A., Vergni D., Vulpiani A.: Thin front propagation in steady and unsteady cellular flows. Phys. Fluids 15(3), 679–688 (2003) · Zbl 1185.76077
[13] Embid P., Majda A., Souganidis P.E.: Comparison of turbulent flame speeds from complete averaging and the G-equation. Phys. Fluids 7(8), 2052–2060 (1995) · Zbl 1039.80504
[14] Evans L.C.: The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111(3–4), 359–375 (1989) · Zbl 0679.35001
[15] Evans L.C., Gariepy R.F.: Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992 · Zbl 0804.28001
[16] Frankowska H.: Hamilton-Jacobi equations: viscosity solutions and generalized gradients. J. Math. Anal. Appl. 141(1), 21–26 (1989) · Zbl 0727.35028
[17] Imbert C., Monneau R.: Homogenization of first-order equations with u/eriodic Hamiltonians. Part I: local equations. Archive for Rational Mechanics and Analysis 187(1), 49–89 (2008) · Zbl 1127.70009
[18] Ishii H., Pires G., Souganidis P.E.: Threshold dynamics type approximation schemes for propagating fronts. J. Math. Soc. Jpn. 51(2), 267–308 (1999) · Zbl 0935.53006
[19] Lions P.-L., Papanicolaou G., Varadhan S.R.S.: Homogenization of Hamilton- Jacovi Equations, unpublished manuscript, c. 1986
[20] Majda A., Souganidis P.E.: Large-scale front dynamics for turbulent reaction- diffusion equations with separated velocity scales. Nonlinearity 7(1), 1–30 (1994) · Zbl 0839.76093
[21] Nolen J., Xin J.: Bounds on front speeds for inviscid and viscous G-equations, preprint, 2009 · Zbl 1256.35090
[22] Oberman A.: Ph.D Thesis, University of Chicago, 2001
[23] Peters N.: Turbulent Combustion. Cambridge University Press, Cambridge (2000) · Zbl 0955.76002
[24] Williams F.A.: Turbulent Combustion, in The Mathematics of Combustion, Buckmaster, J.D. Ed. Society for Industrial and Applied Mathematics, 1985, pp. 97–131
[25] Xin J., Yu Y.: Periodic Homogenization of Inviscid G-Equation for Incompressible Flows, preprint · Zbl 1372.76085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.