Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T00:51:33.764Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential convergence to equilibrium via Lyapounov functionals for reaction-diffusion equations arisingfrom non reversible chemical kinetics

Published online by Cambridge University Press:  05 December 2008

Marzia Bisi ,
Laurent Desvillettes  and
Giampiero Spiga
Marzia Bisi
Affiliation:
Dipartimento di Matematica, Università di Parma, Viale G.P. Usberti 53/A, 43100 Parma, Italy. marzia.bisi@unipr.it giampiero.spiga@unipr.it
Laurent Desvillettes
Affiliation:
CMLA, ENS Cachan, IUF & CNRS, PRES UniverSud, 61 avenue du Président Wilson, 94230 Cachan, France. desville@cmla.ens-cachan.fr
Giampiero Spiga
Affiliation:
Dipartimento di Matematica, Università di Parma, Viale G.P. Usberti 53/A, 43100 Parma, Italy. marzia.bisi@unipr.it giampiero.spiga@unipr.it
Get access

Abstract


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


Keywords

Entropy methodsLyapounov functionalsreaction-diffusion equations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 1 , January 2009 , pp. 151 - 172
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, A., Carrillo, J.A., Desvillettes, L., Dolbeault, J., Jungel, A., Lederman, C., Markowich, P.A., Toscani, G. and Villani, C., Entropies and equilibria of many-particle systems: An essay on recent research. Monat. Mathematik 142 (2004) 35–43. CrossRef
Bisi, M. and Desvillettes, L., From reactive Boltzmann equations to reaction-diffusion systems. J. Stat. Phys. 124 (2006) 881–912. CrossRef
Bisi, M. and Spiga, G., Diatomic gas diffusing in a background medium: kinetic approach and reaction-diffusion equations. Commun. Math. Sci. 4 (2006) 779–798. CrossRef
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).
Cáceres, M., Carrillo, J. and Toscani, G., Long-time behavior for a nonlinear fourth order parabolic equation. Trans. Amer. Math. Soc. 357 (2005) 1161–1175. CrossRef
Carrillo, J.A. and Toscani, G., Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana University Math. J. 49 (2000) 113–142. CrossRef
Del Pino, M. and Dolbeault, J., Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81 (2002) 847–875. CrossRef
Desvillettes, L., About entropy methods for reaction-diffusion equations. Rivista Matematica dell'Università di Parma 7 (2007) 81–123.
Desvillettes, L. and Fellner, K., Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations. J. Math. Anal. Appl. 319 (2006) 157–176. CrossRef
L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion systems: Degenerate diffusion. Discrete Contin. Dyn. Syst. Supplement (2007) 304–312.
L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds. Revista Mat. Iberoamericana (to appear).
Desvillettes, L. and Villani, C., On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications. Comm. Partial Differ. Equ. 25 (2000) 261–298. CrossRef
V. Giovangigli, Multicomponent Flow Modeling. Birkhäuser, Boston (1999).
Groppi, M., Rossani, A. and Spiga, G., Kinetic theory of a diatomic gas with reactions of dissociation and recombination through a transition state. J. Phys. A 33 (2000) 8819–8833. CrossRef
Kirane, M., 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.
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).
Masuda, K., On the global existence and asymptotic behavior of solution of reaction-diffusion equations. Hokkaido Math. J. 12 (1983) 360–370. CrossRef
McLennan, J.A., Boltzmann equation for a dissociating gas. J. Stat. Phys. 57 (1989) 887–905. CrossRef
Y. Sone, Kinetic Theory and Fluid Dynamics. Birkhäuser, Boston (2002).
Toscani, G. and Villani, C., Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Comm. Math. Phys. 203 (1999) 667–706. CrossRef
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.