Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium

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December 2014, 9(4): 669-682. doi: 10.3934/nhm.2014.9.669

Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium

Danielle Hilhorst ^1, and Hideki Murakawa ^2,

1.	Laboratoire de Mathématiques, CNRS et Université de Paris-Sud, 91405 Orsay, France
2.	Faculty of Mathematics, Kyushu University, 744 Motooka, Nishiku, Fukuoka, 819-0395, Japan

Received March 2014 Revised October 2014 Published December 2014

Abstract
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In this paper we consider a three-component reaction-diffusion system with a fast precipitation and dissolution reaction term. We investigate its singular limit as the reaction rate tends to infinity. The limit problem is described by a combination of a Stefan problem and a linear heat equation. The rate of convergence with respect to the reaction rate is established in a specific case.

Keywords: dissolution, heat equation., precipitation, fast reaction limit, singular limit analysis, Stefan problem, Reaction-diffusion system.

Mathematics Subject Classification: 35K05, 35K51, 35K57, 80A22, 80A3.

Citation: Danielle Hilhorst, Hideki Murakawa. Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium. Networks & Heterogeneous Media, 2014, 9 (4) : 669-682. doi: 10.3934/nhm.2014.9.669

References:

[1]	N. Bouillard, R. Eymard, M. Henry, R. Herbin and D. Hilhorst, A fast precipitation and dissolution reaction for a reaction-diffusion system arising in a porous medium,, Nonlinear Anal. Real World Appl., 10 (2009), 629. doi: 10.1016/j.nonrwa.2007.10.019. Google Scholar
[2]	N. Bouillard, R. Eymard, R. Herbin and Ph. Montarnal, Diffusion with dissolution and precipitation in a porous medium: Mathematical analysis and numerical approximation of a simplified model,, Math. Mod. Numer. Anal., 41 (2007), 975. doi: 10.1051/m2an:2007047. Google Scholar
[3]	H. Brézis, Analyse Fonctionnelle,, Masson, (1983). Google Scholar
[4]	R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, handbook of numerical analysis,, Handb. Numer. Anal., VII* (2000), 713. Google Scholar*
[5]	J. Pousin, Infinitely fast kinetics for dissolution and diffusion in open reactive systems,, Nonlinear Analysis, 39 (2000), 261. doi: 10.1016/S0362-546X(98)00162-X. Google Scholar

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References:

[1]	N. Bouillard, R. Eymard, M. Henry, R. Herbin and D. Hilhorst, A fast precipitation and dissolution reaction for a reaction-diffusion system arising in a porous medium,, Nonlinear Anal. Real World Appl., 10 (2009), 629. doi: 10.1016/j.nonrwa.2007.10.019. Google Scholar
[2]	N. Bouillard, R. Eymard, R. Herbin and Ph. Montarnal, Diffusion with dissolution and precipitation in a porous medium: Mathematical analysis and numerical approximation of a simplified model,, Math. Mod. Numer. Anal., 41 (2007), 975. doi: 10.1051/m2an:2007047. Google Scholar
[3]	H. Brézis, Analyse Fonctionnelle,, Masson, (1983). Google Scholar
[4]	R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, handbook of numerical analysis,, Handb. Numer. Anal., VII* (2000), 713. Google Scholar*
[5]	J. Pousin, Infinitely fast kinetics for dissolution and diffusion in open reactive systems,, Nonlinear Analysis, 39 (2000), 261. doi: 10.1016/S0362-546X(98)00162-X. Google Scholar

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