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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

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), 629-638. 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), 975-1000. 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), 713-1020.  Google Scholar

[5]

J. Pousin, Infinitely fast kinetics for dissolution and diffusion in open reactive systems, Nonlinear Analysis, 39 (2000), 261-279. doi: 10.1016/S0362-546X(98)00162-X.  Google Scholar

show all references

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), 629-638. 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), 975-1000. 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), 713-1020.  Google Scholar

[5]

J. Pousin, Infinitely fast kinetics for dissolution and diffusion in open reactive systems, Nonlinear Analysis, 39 (2000), 261-279. doi: 10.1016/S0362-546X(98)00162-X.  Google Scholar

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