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

Abstract

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. Résumé.On s'intéresseà un problèmeà frontière libre bidimensionnel, avec conditions de sautà l'interface. Le problème parabolique admet comme solutions des ondes progressives planes, qui sont orbitalement stables si le paramètre u * ne dépasse pas la valeur u c * . Ce point critique est la limite d'une suite décroissante de points de bifurcation (u k * ). Dans cet article, onétudie la structure des branches bifurquées 2Dà partir de la branche triviale formée des ondes planes, pour k petit. Notre technique consisteàéliminer le front inconnu, pour se ramenerà un problème totalement non linéaireéquivalent, auquel on applique le théorème de bifurcation de Crandall-Rabinowitz pour uneétude locale. La reformulation totalement non linéaire du problème s'avèreégalement bien adaptéeà la mise en oeuvre de méthodes numériques pour le suivi global des branches bifurquées, en particulier par continuation.Mathematics Subject Classification. 35R35, 35B32, 35K55, 65M06, 65F99.

Instability of free interfaces in premixed flame propagation

“…In the paper [8] and related works (see in particular [6,7,10,12,22,23,24]), we presented a method by which the flame front can be eliminated and, mutatis mutandis, the system reformulated as a fully nonlinear problems (see [25]). This new formulation has proved effective for local existence and stability analysis (see above references), and also numerical simulation (see [2]).…”