October  2011, 16(3): 739-766. doi: 10.3934/dcdsb.2011.16.739

Existence of radial stationary solutions for a system in combustion theory

1. 

INRA, Equipe BIOSP, Centre de Recherche d'Avignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9, France

2. 

Departamento de Ingeniería Matemática and CMM (UMI 2807 CNRS), Universidad de Chile, Blanco Encalada 2120 - 5 Piso, Santiago, Chile

Received  June 2010 Revised  February 2011 Published  June 2011

In this paper, we construct radially symmetric solutions of a nonlinear non-cooperative elliptic system derived from a model for flame balls with radiation losses. This model is based on a one step kinetic reaction and our system is obtained by approximating the standard Arrehnius law by an ignition nonlinearity, and by simplifying the term that models radiation. We prove the existence of 2 solutions using degree theory.
Citation: Jérôme Coville, Juan Dávila. Existence of radial stationary solutions for a system in combustion theory. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 739-766. doi: 10.3934/dcdsb.2011.16.739
References:
[1]

R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts," Clarendon Press, Oxford, 1975.

[2]

J. B. van den Berg, V. Guyonne and J. Hulshof, Flame balls for a free boundary combustion model with radiative transfer, SIAM J. Appl. Math., 67 (2006), 116-137. doi: 10.1137/050636516.

[3]

H. Berestycki and B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées, Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Vol. X (Paris, 1987-1988), 65-129, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991.

[4]

H. Berestycki, B. Larrouturou and J.-M. Roquejoffre, Mathematical investigation of the cold boundary difficulty in flame propagation theory, Dynamical Issues in Combustion Theory (Minneapolis, MN, 1989), 37-61, IMA Vol. Math. Appl., 35, Springer, New York, 1991.

[5]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[6]

H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal., 16 (1985), 1207-1242. doi: 10.1137/0516088.

[7]

H. Bockhorn, J. Fröhlich and K. Schneider, An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls, Combustion Theory and Modelling, 3 (1999), 177-198. doi: 10.1088/1364-7830/3/1/010.

[8]

I. Brailovsky and G. I. Sivashinsky, On stationay and travelling flameballs, Combustion and Flame, 110 (1997), 524-529. doi: 10.1016/S0010-2180(97)00001-1.

[9]

J. Brindley, N. A. Jivraj, J. H. Merkin and S. K. Scott, Stationary-state solutions for coupled reaction-diffusion and temperature-conduction equations. II. Spherical geometry with Dirichlet boundary conditions, Proc. Roy. Soc. London Ser. A, 430 (1990), 479-488. doi: 10.1098/rspa.1990.0102.

[10]

J. Buckmaster, G. Joulin and P. Ronney, The structure and stability of nonadiabatic flame balls, Combustion and Flame, 79 (1990), 381-392. doi: 10.1016/0010-2180(90)90147-J.

[11]

J. Buckmaster, G. Joulin and P. D. Ronney, The structure and stability of nonadiabatic flame balls: II. Effects of far-field losses, Combustion and Flame, 84 (1991), 411-422. doi: 10.1016/0010-2180(91)90015-4.

[12]

C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis, SIAM J. Appl. Math., 51 (1991), 1315-1326. doi: 10.1137/0151066.

[13]

R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons Ltd., Chichester, 2003.

[14]

P. Clément and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 97-121.

[15]

Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230. doi: 10.1006/jdeq.2000.3932.

[16]

V. Giovangigli, Nonadiabatic plane laminar flames and their singular limits, SIAM J. Math. Anal., 21 (1990), 1305-1325. doi: 10.1137/0521072.

[17]

L. Glangetas and J.-M. Roquejoffre, Rigorous derivation of the dispersion relation in a combustion model with heat losses, Preprint Publications du Laboratoire d'Analyse Numérique R95032, (1995).

[18]

V. Guyonne and L. Lorenzi, Instability in a flame ball problem, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 315-350.

[19]

V. Guyonne and P. Noble, On a model of flame ball with radiative transfer, SIAM J. Appl. Math., 67 (2007), 854-868. doi: 10.1137/060659612.

[20]

V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behaviour for a competing species problem with diffusion, Dynamical Systems and Applications, World Sci. Ser. Appl. Anal., vol. 4, World Sci. Publ., River Edge, NJ, 1995, pp. 343-358.

[21]

L. Kagan and G. I. Sivashinsky, Self-fragmentation of nonadiabatic cellular flames, Combust. Flame, 108 (1997), 220-226. doi: 10.1016/S0010-2180(96)00108-3.

[22]

L. Kagan, S. Minaev and G. I. Sivashinsky, On self-drifting flame balls, Math. Comput. Simulation, 65 (2004), 511-520. doi: 10.1016/j.matcom.2004.01.013.

[23]

A. K. Kapila, B. J. Matkowsky and J. Vega, Reactive-diffusive system with Arrhenius kinetics: peculiarities of the spherical geometry, SIAM J. Appl. Math., 38 (1980), 382-401. doi: 10.1137/0138032.

[24]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[25]

C. Lederman, J.-M. Roquejoffre and N. Wolanski, Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames, Ann. Mat. Pura Appl., 183 (2004), 173-239. doi: 10.1007/s10231-003-0085-1.

[26]

C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis, SIAM J. Appl. Math., 51 (1991), 1315-1326. doi: 10.1137/0151066.

[27]

B. Lewis and G. von Elbe, "Combustion, Flames and Explosion of Gases," 3rd ed., Academic Press, Orlando, 1987.

[28]

M. Marion, Qualitative properties of a nonlinear system for laminar flames without ignition temperature, Nonlinear Anal., 9 (1985), 1269-1292. doi: 10.1016/0362-546X(85)90035-5.

[29]

S. Minaev, L. Kagan, G. Joulin and G. I. Sivashinsky, On self-drifting flame balls, Combust. Theory Model., 5 (2001), 609-622. doi: 10.1088/1364-7830/5/4/306.

[30]

P. Ronney, Near-Limit Flame Structures at Low Lewis Number, Combustion and Flame, 82 (1990), 1-14. doi: 10.1016/0010-2180(90)90074-2.

[31]

L. Roques, Study of the premixed flame model with heat losses. The existence of two solutions, European J. Appl. Math., 16 (2005), 741-765. doi: 10.1017/S0956792505006431.

[32]

G. Sagon, Steady fronts solutions of semilinear elliptic equations in exterior domains arising in flame propagation, Ann. Mat. Pura Appl., 185 (2006), 273-291.

[33]

A. A. Shah, R. W. Thatcher and J. W. Dold, Stability of a spherical flame ball in a porous medium, Combustion Theory and Modelling, 4 (2000), 511-534. doi: 10.1088/1364-7830/4/4/308.

[34]

Y. B. Zeldovich, A theory of the limit of slow flame propagation, Zh. Prikl. Mekh. i Tekhn. Fiz., 1 (1941), 159-169.

show all references

References:
[1]

R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts," Clarendon Press, Oxford, 1975.

[2]

J. B. van den Berg, V. Guyonne and J. Hulshof, Flame balls for a free boundary combustion model with radiative transfer, SIAM J. Appl. Math., 67 (2006), 116-137. doi: 10.1137/050636516.

[3]

H. Berestycki and B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées, Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Vol. X (Paris, 1987-1988), 65-129, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991.

[4]

H. Berestycki, B. Larrouturou and J.-M. Roquejoffre, Mathematical investigation of the cold boundary difficulty in flame propagation theory, Dynamical Issues in Combustion Theory (Minneapolis, MN, 1989), 37-61, IMA Vol. Math. Appl., 35, Springer, New York, 1991.

[5]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[6]

H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal., 16 (1985), 1207-1242. doi: 10.1137/0516088.

[7]

H. Bockhorn, J. Fröhlich and K. Schneider, An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls, Combustion Theory and Modelling, 3 (1999), 177-198. doi: 10.1088/1364-7830/3/1/010.

[8]

I. Brailovsky and G. I. Sivashinsky, On stationay and travelling flameballs, Combustion and Flame, 110 (1997), 524-529. doi: 10.1016/S0010-2180(97)00001-1.

[9]

J. Brindley, N. A. Jivraj, J. H. Merkin and S. K. Scott, Stationary-state solutions for coupled reaction-diffusion and temperature-conduction equations. II. Spherical geometry with Dirichlet boundary conditions, Proc. Roy. Soc. London Ser. A, 430 (1990), 479-488. doi: 10.1098/rspa.1990.0102.

[10]

J. Buckmaster, G. Joulin and P. Ronney, The structure and stability of nonadiabatic flame balls, Combustion and Flame, 79 (1990), 381-392. doi: 10.1016/0010-2180(90)90147-J.

[11]

J. Buckmaster, G. Joulin and P. D. Ronney, The structure and stability of nonadiabatic flame balls: II. Effects of far-field losses, Combustion and Flame, 84 (1991), 411-422. doi: 10.1016/0010-2180(91)90015-4.

[12]

C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis, SIAM J. Appl. Math., 51 (1991), 1315-1326. doi: 10.1137/0151066.

[13]

R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons Ltd., Chichester, 2003.

[14]

P. Clément and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 97-121.

[15]

Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, J. Differential Equations, 173 (2001), 213-230. doi: 10.1006/jdeq.2000.3932.

[16]

V. Giovangigli, Nonadiabatic plane laminar flames and their singular limits, SIAM J. Math. Anal., 21 (1990), 1305-1325. doi: 10.1137/0521072.

[17]

L. Glangetas and J.-M. Roquejoffre, Rigorous derivation of the dispersion relation in a combustion model with heat losses, Preprint Publications du Laboratoire d'Analyse Numérique R95032, (1995).

[18]

V. Guyonne and L. Lorenzi, Instability in a flame ball problem, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 315-350.

[19]

V. Guyonne and P. Noble, On a model of flame ball with radiative transfer, SIAM J. Appl. Math., 67 (2007), 854-868. doi: 10.1137/060659612.

[20]

V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behaviour for a competing species problem with diffusion, Dynamical Systems and Applications, World Sci. Ser. Appl. Anal., vol. 4, World Sci. Publ., River Edge, NJ, 1995, pp. 343-358.

[21]

L. Kagan and G. I. Sivashinsky, Self-fragmentation of nonadiabatic cellular flames, Combust. Flame, 108 (1997), 220-226. doi: 10.1016/S0010-2180(96)00108-3.

[22]

L. Kagan, S. Minaev and G. I. Sivashinsky, On self-drifting flame balls, Math. Comput. Simulation, 65 (2004), 511-520. doi: 10.1016/j.matcom.2004.01.013.

[23]

A. K. Kapila, B. J. Matkowsky and J. Vega, Reactive-diffusive system with Arrhenius kinetics: peculiarities of the spherical geometry, SIAM J. Appl. Math., 38 (1980), 382-401. doi: 10.1137/0138032.

[24]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[25]

C. Lederman, J.-M. Roquejoffre and N. Wolanski, Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames, Ann. Mat. Pura Appl., 183 (2004), 173-239. doi: 10.1007/s10231-003-0085-1.

[26]

C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis, SIAM J. Appl. Math., 51 (1991), 1315-1326. doi: 10.1137/0151066.

[27]

B. Lewis and G. von Elbe, "Combustion, Flames and Explosion of Gases," 3rd ed., Academic Press, Orlando, 1987.

[28]

M. Marion, Qualitative properties of a nonlinear system for laminar flames without ignition temperature, Nonlinear Anal., 9 (1985), 1269-1292. doi: 10.1016/0362-546X(85)90035-5.

[29]

S. Minaev, L. Kagan, G. Joulin and G. I. Sivashinsky, On self-drifting flame balls, Combust. Theory Model., 5 (2001), 609-622. doi: 10.1088/1364-7830/5/4/306.

[30]

P. Ronney, Near-Limit Flame Structures at Low Lewis Number, Combustion and Flame, 82 (1990), 1-14. doi: 10.1016/0010-2180(90)90074-2.

[31]

L. Roques, Study of the premixed flame model with heat losses. The existence of two solutions, European J. Appl. Math., 16 (2005), 741-765. doi: 10.1017/S0956792505006431.

[32]

G. Sagon, Steady fronts solutions of semilinear elliptic equations in exterior domains arising in flame propagation, Ann. Mat. Pura Appl., 185 (2006), 273-291.

[33]

A. A. Shah, R. W. Thatcher and J. W. Dold, Stability of a spherical flame ball in a porous medium, Combustion Theory and Modelling, 4 (2000), 511-534. doi: 10.1088/1364-7830/4/4/308.

[34]

Y. B. Zeldovich, A theory of the limit of slow flame propagation, Zh. Prikl. Mekh. i Tekhn. Fiz., 1 (1941), 159-169.

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