Computation of bifurcated branches in a free boundary problem arising in combustion theory

Olivier Baconneau; Claude-Michel Brauner; Alessandra Lunardi

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 2, page 223-339
  • ISSN: 0764-583X

Abstract

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We consider a parabolic 2D Free Boundary Problem, with jump conditions at the interface. Its planar travelling-wave solutions are orbitally stable provided the bifurcation parameter u * does not exceed a critical value u * c . The latter is the limit of a decreasing sequence ( u * k ) of bifurcation points. The paper deals with the study of the 2D bifurcated branches from the planar branch, for small k. Our technique is based on the elimination of the unknown front, turning the problem into a fully nonlinear one, to which we can apply the Crandall-Rabinowitz bifurcation theorem for a local study. We point out that the fully nonlinear reformulation of the FBP can also serve to develop efficient numerical schemes in view of global information, such as techniques based on arc length continuation.

How to cite

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Baconneau, Olivier, Brauner, Claude-Michel, and Lunardi, Alessandra. "Computation of bifurcated branches in a free boundary problem arising in combustion theory." ESAIM: Mathematical Modelling and Numerical Analysis 34.2 (2010): 223-339. <http://eudml.org/doc/197537>.

@article{Baconneau2010,
abstract = { We consider a parabolic 2D Free Boundary Problem, with jump conditions at the interface. Its planar travelling-wave solutions are orbitally stable provided the bifurcation parameter $u_*$ does not exceed a critical value $u_\{*\}^\{c\}$. The latter is the limit of a decreasing sequence $(u_\{*\}^\{k\})$ of bifurcation points. The paper deals with the study of the 2D bifurcated branches from the planar branch, for small k. Our technique is based on the elimination of the unknown front, turning the problem into a fully nonlinear one, to which we can apply the Crandall-Rabinowitz bifurcation theorem for a local study. We point out that the fully nonlinear reformulation of the FBP can also serve to develop efficient numerical schemes in view of global information, such as techniques based on arc length continuation. },
author = {Baconneau, Olivier, Brauner, Claude-Michel, Lunardi, Alessandra},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Free Boundary Problem; bifurcation; fully nonlinear equations; continuation.; free boundary problem; travelling-wave solutions; bifurcation points; bifurcated branches; arc length continuation},
language = {eng},
month = {3},
number = {2},
pages = {223-339},
publisher = {EDP Sciences},
title = {Computation of bifurcated branches in a free boundary problem arising in combustion theory},
url = {http://eudml.org/doc/197537},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Baconneau, Olivier
AU - Brauner, Claude-Michel
AU - Lunardi, Alessandra
TI - Computation of bifurcated branches in a free boundary problem arising in combustion theory
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 2
SP - 223
EP - 339
AB - We consider a parabolic 2D Free Boundary Problem, with jump conditions at the interface. Its planar travelling-wave solutions are orbitally stable provided the bifurcation parameter $u_*$ does not exceed a critical value $u_{*}^{c}$. The latter is the limit of a decreasing sequence $(u_{*}^{k})$ of bifurcation points. The paper deals with the study of the 2D bifurcated branches from the planar branch, for small k. Our technique is based on the elimination of the unknown front, turning the problem into a fully nonlinear one, to which we can apply the Crandall-Rabinowitz bifurcation theorem for a local study. We point out that the fully nonlinear reformulation of the FBP can also serve to develop efficient numerical schemes in view of global information, such as techniques based on arc length continuation.
LA - eng
KW - Free Boundary Problem; bifurcation; fully nonlinear equations; continuation.; free boundary problem; travelling-wave solutions; bifurcation points; bifurcated branches; arc length continuation
UR - http://eudml.org/doc/197537
ER -

References

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  1. O. Baconneau, Bifurcation de fronts pour un problème à frontière libre en combustion. Ph.D. thesis, Université Bordeaux 1 (1998).  
  2. C.-M. Brauner, J. Hulshof and A. Lunardi, A general approach to stability in free boundary problems. J. Differential Equations (to appear).  Zbl0973.35200
  3. C.-M. Brauner and A. Lunardi, Bifurcation of nonplanar travelling waves in a free boundary problem. Nonlinear Analysis T. M. A. (to appear).  Zbl0981.35102
  4. C.-M. Brauner, A. Lunardi and Cl. Schmidt-Lainé, Stability of travelling waves with interface conditions. Nonlinear Analysis T. M. A.19 (1992) 465-484.  
  5. C.-M. Brauner, A. Lunardi and Cl. Schmidt-Lainé, Multidimensional stability analysis of planar travelling waves. Appl. Math. Lett.7 (1994) 1-4.  Zbl0814.35148
  6. C.-M. Brauner, A. Lunardi and Cl. Schmidt-Lainé, Stability of travelling waves in a multidimensional free boundary problem. Nonlinear Analysis T.M.A. (to appear).  
  7. M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues. J. Funct. Anal.8 (1971) 321-340.  Zbl0219.46015
  8. H.B. Keller, Numerical solution of bifurcation and non linear eigenvalue problems, P. Rabinowitz Ed., Academic Press, New York (1978) 73-94.  
  9. D.H. Sattinger, Stability of waves of nonlinear parabolic equations. Adv. Math.22 (1976) 141-178.  
  10. D.S. Stewart and G.S.S. Ludford, The acceleration of fast deflagration waves. Z.A.M.M.63 (1983) 291-302.  Zbl0564.76110
  11. J.L. Vazquez, The Free Boundary Problem for the Heat Equation with fixed Gradient Condition, Proc. Int. Conf. "Free Boundary Problem and Applications'', Zakopane, Pitman Res. Notes Math. 363, Longman (1996).  

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