Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A free-boundary problem for the heat equation arising in flame propagation

Authors: Luis A. Caffarelli and Juan L. Vázquez
Journal: Trans. Amer. Math. Soc. 347 (1995), 411-441
MSC: Primary 35K57; Secondary 35R35, 80A22, 80A25
MathSciNet review: 1260199
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a new free-boundary problem for the heat equation, of interest in combustion theory. It is obtained in the description of laminar flames as an asymptotic limit for high activation energy. The problem asks for the determination of a domain in space-time, $ \Omega \subset {{\mathbf{R}}^n} \times (0,T)$, and a function $ u(x,t) \geqslant 0$ defined in $ \Omega $, such that $ {u_t} = \Delta u$ in $ \Omega ,\;u$ takes certain initial conditions, $ u(x,0) = {u_0}(x)$ for $ x \in {\Omega _0} = \partial \Omega \cap \{ t = 0\} $, and two conditions are satisfied on the free boundary $ \Gamma = \partial \Omega \cap \{ t > 0\} :u = 0$ and $ {u_\nu } = - 1$, where $ {u_\nu }$ denotes the derivative of $ u$ along the spatial exterior normal to $ \Gamma $. We approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data.

References [Enhancements On Off] (What's this?)

  • [BCN] H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Uniform estimates for regularization of free boundary problems, Analysis and Partial Differential Equations, Marcel Dekker, New York, 1990. MR 1044809 (91b:35112)
  • [BL] H. Berestycki and B. Larrouturou, A semilinear elliptic equation in a strip arising in a two-dimensional flame propagation model, J. Reine Angew. Math. 396 (1989), 14-40. MR 988546 (91a:35071)
  • [BLL] H. Berestycki, B. Larrouturou, and P. L. Lions, Multi-dimensional travelling-wave solutions of a flame propagation model, Arch. Rational Mech. Anal. 111 (1990), 33-49. MR 1051478 (91h:35148)
  • [BLN] H. Berestycki, B. Larrouturou, and L. Nirenberg, A nonlinear elliptic problem describing the propagation of a curved premixed flame, NATO Advanced Research Workshop on Mathematical Modelling in Combustion and Related Topics.
  • [BHS] M. Bertsch, D. Hilhorst, and C. Schmidt-Lainé, The well-posedness of a free-boundary problem arising in combustion theory, Preprint 21 Dép. Math., E. N. S. Lyon, France, 1989.
  • [BLS] C.-M. Brauner, A. Lunardi, and C. Schmidt-Lainé, Stabilité et instabilité des ondes stationnaires d'un problème de détonation à deux phases, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 697-700. MR 1081628 (92f:80014)
  • [BuL] J. D. Buckmaster and G. S. S. Ludford, Theory of laminar flames, Cambridge Univ. Press, Cambridge, 1982. MR 666866 (84f:80011)
  • [HH] D. Hilhorst and J. Hulshof, An elliptic-parabolic problem in combustion theory: convergence to travelling waves, Proc. Amer. Math. Soc. (to appear). MR 1124123 (92g:35242)
  • [S] D. S. Stewart, Transition to detonation in a model problem, J. Mech. Theor. Appl. 4 (1985), 103-137.
  • [SL] D. S. Stewart and G. S. S. Ludford, Fast deflagration waves, J. Mech. Theor. Appl. 3 (1983), 463-487.
  • [ZBLM] Ya. B. Zel'dovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, Combustion and explosions, Consultants Bureau, New York, 1985. MR 781350 (86b:80014)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K57, 35R35, 80A22, 80A25

Retrieve articles in all journals with MSC: 35K57, 35R35, 80A22, 80A25

Additional Information

Keywords: Heat equation, free-boundary problem, combustion, regularization method, weak solutions
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society