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

DOI:
https://doi.org/10.1090/S0002-9947-00-02663-5

Published electronically:
September 27, 2000

MathSciNet review:
1804512

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 defined in a domain and such that

We also assume that the interior boundary of the positivity set, , so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:

Here 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 . 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.

**[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**, https://doi.org/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**, https://doi.org/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**, https://doi.org/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**, https://doi.org/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**, https://doi.org/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**, https://doi.org/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,*\cyr Lineĭnye i kvazilineĭnye uravneniya parabolicheskogo tipa*, Izdat. “Nauka”, Moscow, 1967 (Russian). MR**0241821**

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

Email:
clederma@dm.uba.ar

**J. L. Vázquez**

Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Email:
juanluis.vazquez@uam.es

**N. Wolanski**

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

Email:
wolanski@dm.uba.ar

DOI:
https://doi.org/10.1090/S0002-9947-00-02663-5

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