Skip to main content
SpringerLink
Log in

Nonuniqueness in a Free Boundary Problem from Combustion

  • Published:
Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We study a parabolic free boundary problem with a fixed gradient condition which serves as a simplified model for the propagation of premixed equidiffusional flames. We give a rigorous justification of an example due to J.L. Vázquez that the initial data in the form of two circular humps leads to the nonuniqueness of limit solutions if the supports of the humps touch at the time of their maximal expansion.

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. Alt, H.W., Caffarelli, L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)

    MATH  MathSciNet  Google Scholar 

  2. Altschuler, S., Angenent, S.B., Giga, Y.: Mean curvature flow through singularities for surfaces of rotation. J. Geom. Anal. 5(3), 293–358 (1995)

    MATH  MathSciNet  Google Scholar 

  3. Baconneau, O., Lunardi, A.: Smooth solutions to a class of free boundary parabolic problems. Trans. Am. Math. Soc. 356(3), 987–1005 (2004) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barles, G., Soner, H.M., Souganidis, P.E.: Front propagation and phase field theory. SIAM J. Control Optim. 31(2), 439–469 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. Buckmaster, J.D., Ludford, G.S.S.: Theory of laminar flames. Cambridge Monographs on Mechanics and Applied Mathematics, Electronic & Electrical Engineering Research Studies: Pattern Recognition & Image Processing Series, vol. 2, p. xii+266. Cambridge University Press, Cambridge (1982),

    Google Scholar 

  6. Caffarelli, L.A., Jerison, D., Kenig C.E.: Global energy minimizers for free boundary problems and full regularity in three dimensions. In: Noncompact problems at the Intersection of Geometry, Analysis, and Topology. Contemp. Math., vol. 350, pp. 83–97. Am. Math. Soc., Providence (2004)

    Google Scholar 

  7. Caffarelli, L.A., Lederman, C., Wolanski, N.: Uniform estimates and limits for a two phase parabolic singular perturbation problem. Indiana Univ. Math. J. 46(2), 453–489 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Caffarelli, L.A., Lederman, C., Wolanski, N.: Pointwise and viscosity solutions for the limit of a two phase parabolic singular perturbation problem. Indiana Univ. Math. J. 46(3), 719–740 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Caffarelli, L.A., Vázquez, J.L.: A free-boundary problem for the heat equation arising in flame propagation. Trans. Am. Math. Soc. 347(2), 411–441 (1995)

    Article  MATH  Google Scholar 

  10. De Silva, D., Jerison, D.: A singular energy minimizing free boundary. Preprint (2005)

  11. Dirr, N., Luckhaus, S., Novaga, M.: A stochastic selection principle in case of fattening for curvature flow. Calc. Var. Partial Differ. Equ. 13(4), 405–425 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature I. J. Differ. Geom. 33(3), 635–681 (1991)

    MATH  MathSciNet  Google Scholar 

  13. Galaktionov, V.A., Hulshof, J., Vazquez, J.L.: Extinction and focusing behaviour of spherical and annular flames described by a free boundary problem. J. Math. Pures Appl. (9) 76(7), 563–608 (1997)

    MATH  MathSciNet  Google Scholar 

  14. Kim, I.: A free boundary problem arising in flame propagation. J. Differ. Equ. 191(2), 470–489 (2003)

    Article  MATH  Google Scholar 

  15. Lederman, C., Vázquez, J.L., Wolanski, N.: Uniqueness of solution to a free boundary problem from combustion. Trans. Am. Math. Soc. 353(2), 655–692 (2001) (electronic)

    Article  MATH  Google Scholar 

  16. Petrosyan, A.: On existence and uniqueness in a free boundary problem from combustion. Commun. Partial Differ. Equ. 27(3–4), 763–789 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Soner, H.M., Souganidis, P.E.: Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature. Commun. Partial Differ. Equ. 18(5–6), 859–894 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Souganidis, P.E., Yip, N.K.: Uniqueness of motion by mean curvature perturbed by stochastic noise, English, with English and French summaries. Ann. Inst. Henri Poincaré Anal. Non Linéaire 21(1), 1–23 (2004)

    MATH  MathSciNet  Google Scholar 

  19. Vázquez, J.L.: The free boundary problem for the heat equation with fixed gradient condition. In: Free Boundary Problems, Theory and Applications, Zakopane, 1995. Pitman Res. Notes Math. Ser., vol. 363, pp. 277–302. Longman, Harlow (1996)

    Google Scholar 

  20. Weiss, G.S.: Partial regularity for a minimum problem with free boundary. J. Geom. Anal. 9(2), 317–326 (1999)

    MATH  MathSciNet  Google Scholar 

  21. Weiss, G.S.: A singular limit arising in combustion theory: fine properties of the free boundary. Calc. Var. Partial Differ. Equ. 17(3), 311–340 (2003)

    MATH  Google Scholar 

  22. Weiss, G.S.: A parabolic free boundary problem with double pinning. Nonlinear Anal. 57(2), 153–172 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zeldovich, Ya.B., Frank-Kamenetski, D.A.: The theory of thermal propagation of flame. Z. Fiz. Him. 12, 100–105 (1938). English transl. in Collected Works of Ya.B. Zeldovich, vol. 1. Princeton Univ. Press (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Arshak Petrosyan & Nung Kwan Yip

Authors
  1. Arshak Petrosyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nung Kwan Yip
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arshak Petrosyan.

Additional information

Communicated by Steven Krantz.

A. Petrosyan was supported in part by NSF grant DMS-0701015.

N.K. Yip was supported in part by NSF grant DMS-0406033.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Petrosyan, A., Yip, N.K. Nonuniqueness in a Free Boundary Problem from Combustion. J Geom Anal 18, 1098–1126 (2008). https://doi.org/10.1007/s12220-008-9044-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-008-9044-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation