American Institute of Mathematical Sciences

Advanced
August  2009, 24(3): 809-826. doi: 10.3934/dcds.2009.24.809

On the stability of high Lewis number combustion fronts

Anna Ghazaryan 1, and  Christopher K. R. T. Jones 1,

1. 

Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States, United States

Received  February 2008 Revised  June 2008 Published  April 2009

We consider wavefronts that arise in a mathematical model for high Lewis number combustion processes. An efficient method for the proof of the existence and uniqueness of combustion fronts is provided by geometric singular perturbation theory. The fronts supported by the model with very large Lewis numbers are small perturbations of the front supported by the model with infinite Lewis number. The question of stability for the fronts is more complicated. Besides discrete spectrum, the system possesses essential spectrum up to the imaginary axis. We show how a geometric approach which involves construction of the Stability Index Bundles can be used to relate the spectral stability of wavefronts with high Lewis numbers to the spectral stability of the front in the case of infinite Lewis number. We discuss the implication for nonlinear stability of fronts with high Lewis numbers. This work builds on the ideas developed by Gardner and Jones [12] and generalized in the papers by Bates, Fife, Gardner and Jones [3, 4].
Keywords: Traveling wave, high Lewis number, slow - fast dynamics, combustion front., stability index.
Mathematics Subject Classification: Primary: 35B35, 80A25; Secondary: 35K57, 34C3.
Citation: Anna Ghazaryan, Christopher K. R. T. Jones. On the stability of high Lewis number combustion fronts. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 809-826. doi: 10.3934/dcds.2009.24.809
[1]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[2]

Zhen-Hui Bu, Zhi-Cheng Wang. Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations Ⅰ. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2395-2430. doi: 10.3934/dcds.2017104
[3]

Debora Amadori, Wen Shen. Front tracking approximations for slow erosion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1481-1502. doi: 10.3934/dcds.2012.32.1481
[4]

Kota Ikeda, Masayasu Mimura. Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 275-305. doi: 10.3934/cpaa.2012.11.275
[5]

Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219
[6]

Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621
[7]

Zhen-Hui Bu, Zhi-Cheng Wang. Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2251-2286. doi: 10.3934/dcds.2018093
[8]

Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, Gregory I. Sivashinsky. A fully nonlinear equation for the flame front in a quasi-steady combustion model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1415-1446. doi: 10.3934/dcds.2010.27.1415
[9]

Luca Biasco, Laura Di Gregorio. Periodic solutions of Birkhoff-Lewis type for the nonlinear wave equation. Conference Publications, 2007, 2007 (Special) : 102-109. doi: 10.3934/proc.2007.2007.102
[10]

C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603
[11]

Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455
[12]

Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507
[13]

Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735
[14]

Jonathan E. Rubin. A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 925-940. doi: 10.3934/dcds.2004.10.925
[15]

Valeria De Mattei. Competition with high number of agents and a major one. Journal of Dynamics & Games, 2016, 3 (4) : 319-334. doi: 10.3934/jdg.2016017
[16]

Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233
[17]

Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123
[18]

Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021
[19]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
[20]

Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 171-215. doi: 10.3934/dcds.2016.36.171

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences