Exponential convergence to equilibrium via Lyapounov functionals for reaction-diffusion equations arising from non reversible chemical kinetics
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 1, p. 151-172

We show that the entropy method, that has been used successfully in order to prove exponential convergence towards equilibrium with explicit constants in many contexts, among which reaction-diffusion systems coming out of reversible chemistry, can also be used when one considers a reaction-diffusion system corresponding to an irreversible mechanism of dissociation/recombination, for which no natural entropy is available.

DOI : https://doi.org/10.1051/m2an:2008045
Classification:  35B40,  35B50,  35K57,  80A30,  92E20
Keywords: entropy methods, Lyapounov functionals, reaction-diffusion equations
@article{M2AN_2009__43_1_151_0,
author = {Bisi, Marzia and Desvillettes, Laurent and Spiga, Giampiero},
title = {Exponential convergence to equilibrium via Lyapounov functionals for reaction-diffusion equations arising from non reversible chemical kinetics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {43},
number = {1},
year = {2009},
pages = {151-172},
doi = {10.1051/m2an:2008045},
zbl = {1155.35312},
mrnumber = {2494798},
language = {en},
url = {http://www.numdam.org/item/M2AN_2009__43_1_151_0}
}

Bisi, Marzia; Desvillettes, Laurent; Spiga, Giampiero. Exponential convergence to equilibrium via Lyapounov functionals for reaction-diffusion equations arising from non reversible chemical kinetics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 1, pp. 151-172. doi : 10.1051/m2an:2008045. http://www.numdam.org/item/M2AN_2009__43_1_151_0/

[1] A. Arnold, J.A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jungel, C. Lederman, P.A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of many-particle systems: An essay on recent research. Monat. Mathematik 142 (2004) 35-43. | MR 2065020 | Zbl 1063.35109

[2] M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems. J. Stat. Phys. 124 (2006) 881-912. | MR 2264629 | Zbl 1134.82323

[3] M. Bisi and G. Spiga, Diatomic gas diffusing in a background medium: kinetic approach and reaction-diffusion equations. Commun. Math. Sci. 4 (2006) 779-798. | MR 2264820 | Zbl 1120.82011

[4] M. Bisi and G. Spiga, Dissociation and recombination of a diatomic gas in a background medium. Proceedings of 25th International Symposium on Rarefied Gas Dynamics (to appear). | MR 2264820

[5] M. Cáceres, J. Carrillo and G. Toscani, Long-time behavior for a nonlinear fourth order parabolic equation. Trans. Amer. Math. Soc. 357 (2005) 1161-1175. | MR 2110436 | Zbl 1077.35028

[6] J.A. Carrillo and G. Toscani, Asymptotic ${L}^{1}$-decay of solutions of the porous medium equation to self-similarity. Indiana University Math. J. 49 (2000) 113-142. | MR 1777035 | Zbl 0963.35098

[7] M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81 (2002) 847-875. | MR 1940370 | Zbl 1112.35310

[8] L. Desvillettes, About entropy methods for reaction-diffusion equations. Rivista Matematica dell'Università di Parma 7 (2007) 81-123. | MR 2375204 | Zbl 1171.35409

[9] L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations. J. Math. Anal. Appl. 319 (2006) 157-176. | MR 2217853 | Zbl 1096.35018

[10] L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion systems: Degenerate diffusion. Discrete Contin. Dyn. Syst. Supplement (2007) 304-312. | MR 2409225 | Zbl 1163.35322

[11] L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds. Revista Mat. Iberoamericana (to appear). | MR 2459198 | Zbl 1171.35330

[12] L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials 25 (2000) 261-298. | MR 1737548 | Zbl 0951.35130

[13] V. Giovangigli, Multicomponent Flow Modeling1999). | MR 1713516 | Zbl 0956.76003

[14] M. Groppi, A. Rossani and G. Spiga, Kinetic theory of a diatomic gas with reactions of dissociation and recombination through a transition state. J. Phys. A 33 (2000) 8819-8833. | MR 1801471 | Zbl 0970.82041

[15] M. Kirane, On stabilization of solutions of the system of parabolic differential equations describing the kinetics of an auto-catalytic reversible chemical reaction. Bull. Institute Math. Academia Sinica 18 (1990) 369-377. | MR 1104954 | Zbl 0731.35056

[16] O.A. Ladyzenskaya, V.A. Solonnikov and N.N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type, Trans. Math. Monographs 23. American Mathematical Society, Providence (1968). | Zbl 0174.15403

[17] K. Masuda, On the global existence and asymptotic behavior of solution of reaction-diffusion equations. Hokkaido Math. J. 12 (1983) 360-370. | MR 719974 | Zbl 0529.35037

[18] J.A. Mclennan, Boltzmann equation for a dissociating gas. J. Stat. Phys. 57 (1989) 887-905.

[19] Y. Sone, Kinetic Theory and Fluid Dynamics2002). | MR 1919070 | Zbl 1021.76002

[20] G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Comm. Math. Phys. 203 (1999) 667-706. | MR 1700142 | Zbl 0944.35066

[21] Y. Yoshizawa, Wave structures of a chemically reacting gas by the kinetic theory of gases, in Rarefied Gas Dynamics, J.L. Potter Ed., A.I.A.A., New York (1977) 501-517.