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Transactions of the American Mathematical Society

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Uniqueness of solution to a free boundary problem from combustion

Authors: C. Lederman, J. L. Vázquez and N. Wolanski
Journal: Trans. Amer. Math. Soc. 353 (2001), 655-692
MSC (1991): Primary 35K05, 35K60, 80A25
Published electronically: September 27, 2000
MathSciNet review: 1804512
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We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function $u(x,t)\geq 0,$ defined in a domain $\mathcal{D} \subset {\mathbb{R}}^{N}\times (0,T)$ and such that

\begin{displaymath}\Delta u+\sum a_{i}\,u_{x_{i}}-u_{t}=0\quad \text{in}\quad \mathcal{D}\cap \{u>0\}. \end{displaymath}

We also assume that the interior boundary of the positivity set, $\mathcal{D} \cap \partial \{u> \nobreak 0\}$, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:

\begin{displaymath}u=0 ,\quad -\partial u/\partial \nu = C. \end{displaymath}

Here $\nu $ denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of $\mathcal{D}$. This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit).

The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.

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  • [AG] Daniele Andreucci and Roberto Gianni, Classical solutions to a multidimensional free boundary problem arising in combustion theory, Comm. Partial Differential Equations 19 (1994), no. 5-6, 803–826. MR 1274541, 10.1080/03605309408821036
  • [BCN] Günther Wildenhain, Approximationseigenschaften der Lösungen elliptischer Differentialgleichungen und die Eindeutigkeitseigenschaft im Kleinen, Complex methods on partial differential equations, Math. Res., vol. 53, Akademie-Verlag, Berlin, 1989, pp. 233–242 (German). MR 1050978
  • [BL] H. Berestycki and B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. X (Paris, 1987–1988) Pitman Res. Notes Math. Ser., vol. 220, Longman Sci. Tech., Harlow, 1991, pp. 65–129 (French, with English summary). MR 1131819
  • [BNS] Henri Berestycki, Basil Nicolaenko, and Bruno Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal. 16 (1985), no. 6, 1207–1242. MR 807905, 10.1137/0516088
  • [BoG] A. Bonnet and L. Glangetas, Non-uniqueness for traveling fronts in the limit of high activation energy, preprint.
  • [BuL] J. D. Buckmaster and G. S. S. Ludford, Theory of laminar flames, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge-New York, 1982. Electronic & Electrical Engineering Research Studies: Pattern Recognition & Image Processing Series, 2. MR 666866
  • [C1] Luis A. Caffarelli, A monotonicity formula for heat functions in disjoint domains, Boundary value problems for partial differential equations and applications, RMA Res. Notes Appl. Math., vol. 29, Masson, Paris, 1993, pp. 53–60. MR 1260438
  • [C2] Luis A. Caffarelli, Uniform Lipschitz regularity of a singular perturbation problem, Differential Integral Equations 8 (1995), no. 7, 1585–1590. MR 1347971
  • [CK] Luis A. Caffarelli and Carlos E. Kenig, Gradient estimates for variable coefficient parabolic equations and singular perturbation problems, Amer. J. Math. 120 (1998), no. 2, 391–439. MR 1613650
  • [CLW1] L. A. Caffarelli, C. Lederman, and N. Wolanski, Uniform estimates and limits for a two phase parabolic singular perturbation problem, Indiana Univ. Math. J. 46 (1997), no. 2, 453–489. MR 1481599, 10.1512/iumj.1997.46.1470
  • [CLW2] L. A. Caffarelli, C. Lederman, and N. Wolanski, Pointwise and viscosity solutions for the limit of a two phase parabolic singular perturbation problem, Indiana Univ. Math. J. 46 (1997), no. 3, 719–740. MR 1488334, 10.1512/iumj.1997.46.1413
  • [CV] Luis A. Caffarelli and Juan L. Vázquez, A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347 (1995), no. 2, 411–441. MR 1260199, 10.1090/S0002-9947-1995-1260199-7
  • [GHV] V.A. Galaktionov, J. Hulshof, and J.L. Vazquez, Extinction and focusing behaviour of spherical and annular flames described by a free boundary problem, Jour. Math. Pure Appl. 76 (1997), 563-608. MR 98h:35328
  • [Gl] L. Glangetas, Étude d’une limite singulière d’un modèle intervenant en combustion, Asymptotic Anal. 5 (1992), no. 4, 317–342 (French, with English summary). MR 1157237
  • [H] Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
  • [HH] D. Hilhorst and J. Hulshof, An elliptic-parabolic problem in combustion theory: convergence to travelling waves, Nonlinear Anal. 17 (1991), no. 6, 519–546. MR 1124123, 10.1016/0362-546X(91)90062-6
  • [KH] L.I. Kamynin and B.N. Himcenko, On applications of the maximum principle to parabolic equations of second order, Soviet Math. Doklady 13(3) (1972), 683-686.
  • [LSU] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). MR 0241822
  • [LVW] C. Lederman, J.L. Vazquez, and N. Wolanski, A mixed semilinear parabolic problem in a noncylindrical space-time domain, Diff. Int. Eqs. (to appear).
  • [LW] Claudia Lederman and Noemi Wolanski, Viscosity solutions and regularity of the free boundary for the limit of an elliptic two phase singular perturbation problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), no. 2, 253–288 (1999). MR 1664689
  • [M] A.M. Meirmanov, On a free boundary problem for parabolic equations, Matem. Sbornik 115 (1981), 532-543 (in Russian); English translation: Math. USSR Sbornik, 43 (1982), 473-484.
  • [V] J. L. Vazquez, The free boundary problem for the heat equation with fixed gradient condition, Free boundary problems, theory and applications (Zakopane, 1995) Pitman Res. Notes Math. Ser., vol. 363, Longman, Harlow, 1996, pp. 277–302. MR 1462990
  • [Ve] Ventsel', A free boundary-value problem for the heat equation, Dokl. Akad. Nauk SSSR 131 (1960), 1000-1003 English translation: Soviet Math. Dokl., 1 (1960).
  • [W] F.A. Williams, Combustion Theory, 2nd. ed., Benjamin-Cummnings, Menlo Park, CA, 1985.
  • [ZF] Ya.B. Zeldovich and D.A. Frank-Kamenetski, The theory of thermal propagation of flames, Zh. Fiz. Khim. 12 (1938), 100-105 (in Russian); English translation in ``Collected Works of Ya.B. Zeldovich", vol. 1, Princeton Univ. Press, 1992.

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Additional Information

C. Lederman
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428), Buenos Aires, Argentina

J. L. Vázquez
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

N. Wolanski
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428), Buenos Aires, Argentina

Keywords: Free-boundary problem, combustion, heat equation, uniqueness, classical solution, viscosity solution, limit solution
Received by editor(s): April 2, 1999
Published electronically: September 27, 2000
Additional Notes: The first and third authors were partially supported by UBA grants EX071, TX47 and grant BID802/OC-AR PICT 03-00000-00137. They are members of CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas of Argentina).
The second author was partially supported by DGICYT Project PB94-0153 and HCM contract FMRX-CT98-0201.
Article copyright: © Copyright 2000 American Mathematical Society