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Traveling Fronts for the Thermo-Diffusive System with Arbitrary Lewis Numbers

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Abstract

We consider KPP-type systems in a cylinder with an arbitrary Lewis number (the ratio of thermal and material diffusivities) in the presence of a shear flow. We show that traveling fronts solutions exist for all Lewis numbers and approach uniform limits at the two ends of the cylinder.

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References

  1. Bagès, M.: Dynamique non triviale en temps grand pour une équation de type KPP en milieu périodique. Ph.D. Thesis, 2007

  2. Berestycki H., Hamel F.: Front propagation in periodic excitable media. Comm. Pure Appl. Math. 55, 949–1032 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berestycki H., Hamel F., Kiselev A., Ryzhik L.: Quenching and propagation in KPP reaction-diffusion equations with a heat loss. Arch. Ration. Mech. Anal. 178, 57–80 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berestycki, H., Larrouturou, B., Lions, P.-L., Roquejoffre, J.-M.: An elliptic system modelling the propagation of a multidimensional flame. Unpublished manuscript, 1995

  5. Berestycki H., Nicolaenko B., Scheurer B.: Travelling wave solutions to combustion models and their singular limits. SIAM J. Math. Anal. 16, 1207–1242 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berestycki H., Nirenberg L.: On the method of moving planes and the sliding method. Bol. Soc. Bras. Mat. 22, 1–37 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Berestycki H., Nirenberg L.: Travelling fronts in cylinders. Ann. Inst. H. Poincaré, Analyse Non Linéaire 9, 497–572 (1992)

    MATH  MathSciNet  Google Scholar 

  8. Billingham J., Needham D.: The development of travelling waves in a quadratic and cubic autocatalysis with unequal diffusion. I. Permanent form travelling waves. Phil. Trans. R. Soc. Lond. A 334, 1–24 (1991)

    MATH  MathSciNet  ADS  Google Scholar 

  9. Bonnet A.: Non-uniqueness for flame propagations when Lewis number is less than 1. Eur. J. Appl. Math. 6, 287–306 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bonnet A., Larrouturou B., Sainsaulieu L.: Numerical stability of multiple planar travelling fronts when Lewis number is less than 1. Phys. D 69, 345–352 (1993)

    Article  MATH  Google Scholar 

  11. Collet P., Xin J.: Global existence and large time asymptotic bounds of L ∞ solutions of thermal diffusive combustion systems on \({\mathbb{R}^n}\) . Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23(4), 625–642 (1996)

    MATH  MathSciNet  Google Scholar 

  12. Ducrot A.: Multi-dimensional combustion waves for Lewis number close to one. Math. Methods Appl. Sci. 30, 291–304 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Ducrot A., Marion M.: Two-dimensional travelling wave solutions of a system modelling near equidiffusional flames. Nonlinear Anal. 61, 1105–1134 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fisher R.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)

    Google Scholar 

  15. Glangetas L., Roquejoffre J.-M.: Bifurcations of travelling waves in the thermodiffusive model for flame propagation. Arch. Ration. Mech. Anal. 134, 341–402 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hamel F., Ryzhik L.: Non-adiabatic KPP fronts with an arbitrary Lewis number. Nonlinearity 18, 2881–2902 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Kolmogorov A.N., Petrovskii I.G., Piskunov N.S.: Étude de l’équation de la chaleurde matière et son application à un problème biologique. Bull. Moskov. Gos. Univ. Math. Mekh. 1, 1–25 (1937)

    Google Scholar 

  18. Marion M.: Qualitative properties of nonlinear system for laminar flames without ignition temperature. Nonlinear Anal. Theory Methods Appl. 9, 1269–1292 (1985)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Metcalf M.J., Merkin J.H., Scott S.K.: Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis. Proc. R. Soc. Lond. A 447, 155–174 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Sivashinsky G.I.: Nonlinear analysis of hydrodynamic instability in laminar flames. I. Derivation of basic equations. Acta Astronaut. 4, 1177–1206 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sivashinsky, G.: Personal communication, 2008

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Authors and Affiliations

  1. LATP, Faculté des Sciences et Techniques, Aix-Marseille Université, Avenue Escadrille Normandie-Niemen, 13397, Marseille Cedex 20, France

    François Hamel

  2. Helmholtz Zentrum München, Institut für Biomathematik und Biometrie, Ingolstädter Landstrasse 1, 85764, Neuherberg, Germany

    François Hamel

  3. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Lenya Ryzhik

Authors
  1. François Hamel
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  2. Lenya Ryzhik
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Correspondence to François Hamel.

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Communicated by P. Rabinowitz

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Hamel, F., Ryzhik, L. Traveling Fronts for the Thermo-Diffusive System with Arbitrary Lewis Numbers. Arch Rational Mech Anal 195, 923–952 (2010). https://doi.org/10.1007/s00205-009-0234-9

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  • DOI: https://doi.org/10.1007/s00205-009-0234-9

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