Geodesic
Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface
Brauner, Claude-Michel1, 2 ; Lorenzi, Luca3 ; Zhang, Mingmin1
1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China
2 Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence Cedex, France
3 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Plesso di Matematica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 581-604

Voir la notice de l'article provenant de la source Numdam

In this paper we analyze the stability of the traveling wave solution for an ignition-temperature first-order reaction model of diffusional-thermal combustion in the case of high Lewis numbers (Le>1). In contrast to conventional Arrhenius kinetics where the reaction zone is infinitely thin, the reaction zone for stepwise temperature kinetics is of order unity. The system of two parabolic PDEs is characterized by a free interface at which ignition temperature Θi is reached. We turn the model to a fully nonlinear problem in a fixed domain. When the Lewis number is large, we define a bifurcation parameter m=Θi/(1−Θi) and a perturbation parameter ε=1/Le. The main result is the existence of a critical value mc(ε) close to mc=6 at which Hopf bifurcation holds for ε small enough. Proofs combine spectral analysis and non-standard application of Hurwitz's Theorem with asymptotics as ε→0.

DOI : 10.1016/j.anihpc.2020.01.002
Classification : 35R35, 35B35, 35K55, 80A25
Keywords: Free interface problems, Traveling wave solutions, Fully nonlinear parabolic systems, Stability, Hopf bifurcation, Combustion

Brauner, Claude-Michel 1, 2 ; Lorenzi, Luca 3 ; Zhang, Mingmin 1

1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China
2 Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence Cedex, France
3 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Plesso di Matematica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy 
@article{AIHPC_2020__37_3_581_0,
     author = {Brauner, Claude-Michel and Lorenzi, Luca and Zhang, Mingmin},
     title = {Stability analysis and {Hopf} bifurcation at high {Lewis} number in a combustion model with free interface},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {581--604},
     publisher = {Elsevier},
     volume = {37},
     number = {3},
     year = {2020},
     doi = {10.1016/j.anihpc.2020.01.002},
     mrnumber = {4093622},
     zbl = {1442.35555},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.01.002/}
}
TY  - JOUR
AU  - Brauner, Claude-Michel
AU  - Lorenzi, Luca
AU  - Zhang, Mingmin
TI  - Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2020
SP  - 581
EP  - 604
VL  - 37
IS  - 3
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.01.002/
DO  - 10.1016/j.anihpc.2020.01.002
LA  - en
ID  - AIHPC_2020__37_3_581_0
ER  - 
%0 Journal Article
%A Brauner, Claude-Michel
%A Lorenzi, Luca
%A Zhang, Mingmin
%T Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface
%J Annales de l'I.H.P. Analyse non linéaire
%D 2020
%P 581-604
%V 37
%N 3
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.01.002/
%R 10.1016/j.anihpc.2020.01.002
%G en
%F AIHPC_2020__37_3_581_0
Brauner, Claude-Michel; Lorenzi, Luca; Zhang, Mingmin. Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 581-604. doi: 10.1016/j.anihpc.2020.01.002

Cité par Sources :