Skip to main content
SpringerLink
Log in

On a strongly damped wave equation for the flame front

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

In two-dimensional free-interface problems, the front dynamics can be modeled by single parabolic equations such as the Kuramoto-Sivashinsky equation (K-S). However, away from the stability threshold, the structure of the front equation may be more involved. In this paper, a generalized K-S equation, a nonlinear wave equation with a strong damping operator, is considered. As a consequence, the associated semigroup turns out to be analytic. Asymptotic convergence to K-S is shown, while numerical results illustrate the dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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.

    MATH  MathSciNet  Google Scholar 

  2. Brauner, C.-M., Frankel, M. L., Hulshof, J. and Sivashinsky, G. I., Weakly nonlinear asymptotic of the κ-θ model of cellular flames: the Q-S equation, Interfaces Free Bound., 7, 2005, 131–146.

    Article  MATH  MathSciNet  Google Scholar 

  3. 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. (Volume dedicated to the memory of Basil Nicolaenko.)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brauner, C.-M., Hulshof, J. and Lorenzi, L., Rigorous derivation of the Kuramoto-Sivashinsky equation in a 2D weakly nonlinear Stefan problem, submitted.

  5. 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.

    Article  MATH  MathSciNet  Google Scholar 

  6. Brauner, C.-M. and Lunardi, A., Instabilities in a two-dimensional combustion model with free boundary, Arch. Ration. Mech. Anal., 154, 2000, 157–182.

    Article  MATH  MathSciNet  Google Scholar 

  7. Carvalho, A. N. and Cholewa, J. W., Strongly damped wave equations in W 1,p(Ω) × L p(Ω), Discrete Contin. Dyn. Syst., Supplement, 2007, 230–239.

  8. Frankel, M. L., Gordon, P. and Sivashinsky, G. I., A stretch-temperature model for flame-flow interaction, Physics Letters A, 361, 2007, 356–359.

    Article  MATH  Google Scholar 

  9. Henry, D., Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, New York, 1981.

    MATH  Google Scholar 

  10. Hyman, J. M. and Nicolaenko, B., The Kuramoto-Sivashinsky equation: a bridge between PDEs and dynamical systems, Phys. D, 18, 1986, 113–126.

    Article  MATH  MathSciNet  Google Scholar 

  11. Lorenzi, L., Regularity and analyticity in a two-dimensional combustion model, Adv. Differential Equations, 7, 2002, 1343–1376.

    MATH  MathSciNet  Google Scholar 

  12. Lorenzi, L., A free boundary problem stemmed from combustion theory, I, Existence, uniqueness and regularity results, J. Math. Anal. Appl., 274, 2002, 505–535.

    Article  MATH  MathSciNet  Google Scholar 

  13. Lorenzi, L., A free boundary problem stemmed from combustion theory, II, Stability, instability and bifurcation results, J. Math. Anal. Appl., 275, 2002, 131–160.

    Article  MATH  MathSciNet  Google Scholar 

  14. Lorenzi, L., Bifurcation of codimension two in a combustion model, Adv. Math. Sci. Appl., 14, 2004, 483–512.

    MATH  MathSciNet  Google Scholar 

  15. Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, Birkhäuser, Basel, 1995.

    MATH  Google Scholar 

  16. 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.

    Article  MATH  MathSciNet  Google Scholar 

  17. Murray, J. D., Mathematical Biology, Biomathematics Texts, 2nd edition, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  18. Sivashinsky, G. I., Nonlinear analysis of hydrodynamic instability in laminar flames, Part 1, Derivation of basic equations, Acta Astronaut., 4, 1977, 1177–1206.

    Article  MATH  MathSciNet  Google Scholar 

  19. Sivashinsky, G. I., On flame propagation under conditions of stoichiometry, SIAM J. Appl. Math., 39, 1980, 67–82.

    Article  MATH  MathSciNet  Google Scholar 

  20. Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, 2nd edition, Springer, Berlin, Heidelberg, New York, 1997.

    Google Scholar 

  21. Turing, A. M., The chemical basis of morphogenesis, Phil. Transl. Roy. Soc. Lond., B237, 1952, 37–72.

    Article  Google Scholar 

  22. Xu, C. J. and Tang, T., Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44(4), 2006, 1759–1779. DOI: 10.1007/s11401-010-0617-0

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, 361005, Xiamen, Fujian, China

    Claude-Michel Brauner

  2. Dipartimento di Matematica, Università degli Studi di Parma, Viale Parco Area delle Scienze 53/A, 43124, Parma, Italy

    Luca Lorenzi

  3. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Gregory I. Sivashinsky

  4. School of Mathematical Sciences, Xiamen University, 361005, Xiamen, Fujian, China

    Chuanju Xu

Authors
  1. Claude-Michel Brauner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luca Lorenzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gregory I. Sivashinsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chuanju Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chuanju Xu.

Additional information

Dedicated to Roger Temam on his 70th Birthday, with Respect and Admiration

Project supported by the National Natural Science Foundation of China (No. 11071203), the 973 High Performance Scientific Computation Research Program (No. 2005CB321703), the US-Israel Binational Science Foundation (No. 2006-151) and the Israel Science Foundation (No. 32/09).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brauner, CM., Lorenzi, L., Sivashinsky, G.I. et al. On a strongly damped wave equation for the flame front. Chin. Ann. Math. Ser. B 31, 819–840 (2010). https://doi.org/10.1007/s11401-010-0616-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-010-0616-1

Keywords

2000 MR Subject Classification

Search

Navigation