Stability of travelling waves in a model for conical flames in two space dimensions

Hamel, François; Monneau, Régis; Roquejoffre, Jean-Michel

doi:10.1016/j.ansens.2004.03.001

Hamel, François ; Monneau, Régis ; Roquejoffre, Jean-Michel

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 37 (2004) no. 3, pp. 469-506.

MR Zbl | 3 citations dans Numdam

DOI : 10.1016/j.ansens.2004.03.001

@article{ASENS_2004_4_37_3_469_0,
     author = {Hamel, Fran\c{c}ois and Monneau, R\'egis and Roquejoffre, Jean-Michel},
     title = {Stability of travelling waves in a model for conical flames in two space dimensions},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {469--506},
     publisher = {Elsevier},
     volume = {Ser. 4, 37},
     number = {3},
     year = {2004},
     doi = {10.1016/j.ansens.2004.03.001},
     mrnumber = {2060484},
     zbl = {1085.35075},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.ansens.2004.03.001/}
}

TY  - JOUR
AU  - Hamel, François
AU  - Monneau, Régis
AU  - Roquejoffre, Jean-Michel
TI  - Stability of travelling waves in a model for conical flames in two space dimensions
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2004
SP  - 469
EP  - 506
VL  - 37
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.ansens.2004.03.001/
DO  - 10.1016/j.ansens.2004.03.001
LA  - en
ID  - ASENS_2004_4_37_3_469_0
ER  -

%0 Journal Article
%A Hamel, François
%A Monneau, Régis
%A Roquejoffre, Jean-Michel
%T Stability of travelling waves in a model for conical flames in two space dimensions
%J Annales scientifiques de l'École Normale Supérieure
%D 2004
%P 469-506
%V 37
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.ansens.2004.03.001/
%R 10.1016/j.ansens.2004.03.001
%G en
%F ASENS_2004_4_37_3_469_0

Hamel, François; Monneau, Régis; Roquejoffre, Jean-Michel. Stability of travelling waves in a model for conical flames in two space dimensions. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 37 (2004) no. 3, pp. 469-506. doi : 10.1016/j.ansens.2004.03.001. http://www.numdam.org/articles/10.1016/j.ansens.2004.03.001/

Bibliographie
Cité par

[1] Agmon S, Nirenberg L, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math. 16 (1963) 121-239. | MR | Zbl

[2] Aronson D.G, Weinberger H.F, Nonlinear diffusion in population genetics, combustion and nerve propagation, in: Part. Diff. Eq. and Related Topics, Lecture Notes in Math., vol. 446, Springer, New York, 1975, pp. 5-49. | MR | Zbl

[3] Berestycki H, Hamel F, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 (2002) 949-1032. | MR | Zbl

[4] Berestycki H, Larrouturou B, Lions P.-L, Multidimensional travelling-wave solutions of a flame propagation model, Arch. Rat. Mech. Anal. 111 (1990) 33-49. | MR | Zbl

[5] Berestycki H, Larrouturou B, Roquejoffre J.-M, Stability of travelling fronts in a curved flame model, Part I, Linear analysis, Arch. Rat. Mech. Anal. 117 (1992) 97-117. | MR | Zbl

[6] Berestycki H, Nicolaenko B, Scheurer B, Traveling waves solutions to combustion models and their singular limits, SIAM J. Math. Anal. 16 (6) (1985) 1207-1242. | MR | Zbl

[7] Berestycki H, Nirenberg L, On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat. 22 (1991) 1-37. | MR | Zbl

[8] Berestycki H, Nirenberg L, Travelling fronts in cylinders, Ann. Inst. H. Poincaré, Anal. Non Lin. 9 (1992) 497-572. | Numdam | MR | Zbl

[9] Bonnet A, Hamel F, Existence of non-planar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal. 31 (1999) 80-118. | MR | Zbl

[10] Bramson M, Convergence of Solutions of the Kolmogorov Equation to Travelling Waves, in: Mem. Amer. Math. Soc., vol. 44, 1983. | MR | Zbl

[11] Brazhnik P.K, Tyson J.J, On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math. 60 (2000) 371-391. | MR | Zbl

[12] Buckmaster J.D, Ludford G.S.S, Lectures on Mathematical Combustion, in: CBMS-NSF Conf. Series in Applied Math., vol. 43, SIAM, 1983. | MR | Zbl

[13] Caffarelli L.A, Cabré X, Fully Nonlinear Elliptic Equations, in: Colloquium Publications, Amer. Math. Soc., vol. 43, 1995. | MR | Zbl

[14] Crandall M.G, Ishii H, Lions P.-L, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1-67. | MR | Zbl

[15] Collet P, Eckmann J.-P, Space-time behaviour in problems of hydrodynamic type: a case study, Nonlinearity 5 (6) (1992) 1265-1302. | MR | Zbl

[16] Fife P.C, Dynamics of internal layers and diffusive interfaces, in: Cbms-Nsf Regional Conference, Series in Applied Mathematics, vol. 53, 1988. | MR | Zbl

[17] Fife P.C, Mcleod J.B, The approach of solutions of non-linear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal. 65 (1977) 335-361. | MR | Zbl

[18] Gilbarg D, Trudinger N.S, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1997. | Zbl

[19] Hamel F, Monneau R, Solutions of semilinear elliptic equations in R^N with conical-shaped level sets, Comm. Partial Differential Equations 25 (2000) 769-819. | MR | Zbl

[20] Hamel F, Monneau R, Existence and uniqueness for a free boundary problem arising in combustion theory, Interfaces Free Boundaries 4 (2002) 167-210. | MR | Zbl

[21] Hamel F, Nadirashvili N, Travelling waves and entire solutions of the Fisher-KPP equation in R^N, Arch. Ration. Mech. Anal. 157 (2001) 91-163. | MR | Zbl

[22] Henry D, Geometric Theory of Semilinear Parabolic Equations, in: Lecture Notes in Math., Springer Verlag, New York, 1981. | MR | Zbl

[23] Ishii H, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE's, Comm. Pure Appl. Math. 42 (1989) 15-45. | MR | Zbl

[24] Joulin G, Dynamique des fronts de flammes, in: Modélisation de la combustion, Images des Mathématiques, CNRS, 1996.

[25] Kanel' Ya.I, Certain problems of burning-theory equations, Sov. Math. Dokl. 2 (1961) 48-51. | MR | Zbl

[26] Kanel' Ya.I, Stabilization of solution of the Cauchy problem for equations encountered in combustion theory, Mat. Sbornik 59 (1962) 245-288. | MR

[27] Levermore C.D, Xin J.X, Multidimensional stability of travelling waves in a bistable reaction-diffusion equation, II, Comm. Partial Differential Equations 17 (1992) 1901-1924. | MR | Zbl

[28] Lewis B, Von Elbe G, Combustion, Flames and Explosions of Gases, Academic Press, New York, 1961.

[29] Mallordy J.-F, Roquejoffre J.-M, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal. 26 (1995) 1-20. | MR | Zbl

[30] Matkowsky B.J, Olagunju D.O, Pulsations in a burner-stabilized premixed plane flame, SIAM J. Appl. Math. 40 (1981) 551-562. | MR | Zbl

[31] Michelson D, Stability of the Bunsen flame profiles in the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal. 27 (1996) 765-781. | MR | Zbl

[32] Ninomiya H., Taniguchi M., Stability of traveling curved fronts in a curvature flow with driving force, Meth. Appl. Anal., submitted for publication. | MR | Zbl

[33] Roquejoffre J.-M, Stability of travelling fronts in a curved flame model, Part II: Non-linear orbital stability, Arch. Rat. Mech. Anal. 117 (1992) 119-153. | MR | Zbl

[34] Roquejoffre J.-M, Convergence to travelling waves for solutions of a class of semilinear parabolic equation, J. Differential Equations 108 (1994) 262-295. | MR | Zbl

[35] Roquejoffre J.-M, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincaré, Anal. Non Lin. 14 (1997) 499-552. | Numdam | MR | Zbl

[36] Sattinger D.H, Stability of waves of nonlinear parabolic systems, Adv. Math. 22 (1976) 312-355. | MR | Zbl

[37] Sattinger D.H, Weighted norms for the stability of travelling waves, J. Differential Equations 25 (1977) 130-144. | MR | Zbl

[38] Sivashinsky G.I, The structure of Bunsen flames, J. Chem. Phys. 62 (1975) 638-643.

[39] Stewart H.B, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980) 299-310. | MR | Zbl

[40] Williams F, Combustion Theory, Addison-Wesley, Reading, MA, 1983.

[41] Xin J.X, Muldimensional stability of travelling waves in a bistable reaction-diffusion equation, I, Comm. Partial Differential Equations 17 (1992) 1889-1899. | MR | Zbl

Cité par Sources :