Abstract
The authors consider a free interface problem which stems from a gas-solid model in combustion with pattern formation. A third-order, fully nonlinear, self-consistent equation for the flame front is derived. Asymptotic methods reveal that the interface approaches a solution to the Kuramoto-Sivashinsky equation. Numerical results which illustrate the dynamics are presented.
References
Berestycki, H., Brauner, C.-M., Clavin, P., et al., Modélisation de la Combustion, Images des Mathématiques, Special Issue, CNRS, Paris, 1996
Brauner, C.-M., Frankel, M. L., Hulshof, J., et al., On the κ-θ model of cellular flames: existence in the large and asymptotics, Discrete Contin. Dyn. Syst. Ser. S, 1, 2008, 27–39.
Brauner, C.-M., Frankel, M. L., Hulshof, J. and Sivashinsky, G. I., Weakly nonlinear asymptotics of the κ-θ model of cellular flames: the Q-S equation, Interfaces Free Bound., 7, 2005, 131–146.
Brauner, C.-M., Hulshof, J. and Lorenzi, L., Stability of the travelling wave in a 2D weakly nonlinear Stefan problem, Kinetic Related Models, 2, 2009, 109–134.
Brauner, C.-M., Hulshof, J. and Lorenzi, L., Rigorous derivation of the Kuramoto-Sivashinsky equation in a 2D weakly nonlinear Stefan problem, Interfaces Free Bound., 13, 2011, 73–103.
Brauner, C.-M., Hulshof, J., Lorenzi, L. and Sivashinsky, G. I., A fully nonlinear equation for the flame front in a quasi-steady combustion model, Discrete Contin. Dyn. Syst. Ser. A, 27, 2010, 1415–1446.
Brauner, C.-M., Lorenzi, L., Sivashinsky, G. I. and Xu, C.-J., On a strongly damped wave equation for the flame front, Chin. Ann. Math., 31B(6), 2010, 819–840.
Brauner, C.-M. and Lunardi, A., Instabilities in a two-dimensional combustion model with free boundary, Arch. Ration. Mech. Anal., 154, 2000, 157–182.
Buckmaster, J. D. and Ludford, G. S. S., Theory of Laminar Flames, Cambridge, New York, 1982.
Eckhaus, W., Asymptotic Analysis of Singular Perturbations, Studies in Mathematics and Its Applications, 9, North-Holland, Amsterdam, New York, 1979.
Haase, M., The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169, Birkhäuser-Verlag, Basel, 2006.
Hyman, J. M. and Nicolaenko, B., The Kuramoto-Sivashinsky equation: a bridge between PDEs and dynamical systems, Phys. D, 18, 1986, 113–126.
Kagan, L. and Sivashinsky, G. I., Pattern formation in flame spread over thin solid fuels, Combust. Theor. Model., 12, 2008, 269–281.
Lions, J.-L., Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Lect. Notes in Math., 323, Springer-Verlag, Berlin, New York, 1970.
Lorenzi, L., Regularity and analyticity in a two-dimensional combustion model, Adv. Diff. Eq., 7, 2002, 1343–1376.
Lorenzi, L., A free boundary problem stemmed from combustion theory. I, existence, uniqueness and regularity results, J. Math. Anal. Appl., 274, 2002, 505–535.
Lorenzi, L., A free boundary problem stemmed from combustion theory. II, stability, instability and bifurcation results, J. Math. Anal. Appl., 275, 2002, 131–160.
Lorenzi, L., Bifurcation of codimension two in a combustion model, Adv. Math. Sci. Appl., 14, 2004, 483–512.
Lorenzi, L. and Lunardi A., Stability in a two-dimensional free boundary combustion model, Nonlinear Anal. 53, 2003, 227–276.
Lorenzi, L. and Lunardi, A., Erratum: “Stability in a two-dimensional free boundary combustion model” [Nonlinear Anal. 53(2003), no. 2, 227–276; MR1959814], Nonlinear Anal., 53(6), 2003, 859–860.
Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.
Matkowsky, B. J. and Sivashinsky, G. I., An asymptotic derivation of two models in flame theory associated with the constant density approximation, SIAM J. Appl. Math., 37, 1979, 686–699.
Sivashinsky, G. I., On flame propagation under conditions of stoichiometry, SIAM J. Appl. Math., 39, 1980, 67–82.
Sivashinsky, G. I., Instabilities, pattern formation and turbulence in flames, Ann. Rev. Fluid Mech., 15, 1983, 179–199.
Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, 2nd edition, Springer-Verlag, New York, 1997.
Zik, O. and Moses, E., Fingering instability in combustion: an extended view, Phys. Rev. E, 60, 1999, 518–531.
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In honor of the scientific heritage of Jacques-Louis Lions
Project supported by a grant from the Fujian Administration of Foreign Expert Affairs, China (No. SZ2011008).
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Brauner, CM., Hu, L. & Lorenzi, L. Asymptotic Analysis in a Gas-Solid Combustion Model with Pattern Formation. Chin. Ann. Math. Ser. B 34, 65–88 (2013). https://doi.org/10.1007/s11401-012-0758-4
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DOI: https://doi.org/10.1007/s11401-012-0758-4