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

Danielle Hilhorst; Hideki Murakawa; Danielle Hilhorst; Hideki Murakawa

doi:10.3934/nhm.2014.9.669

Networks and Heterogeneous Media

2014, Volume 9, Issue 4: 669-682. doi: 10.3934/nhm.2014.9.669

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

Danielle Hilhorst ^{1
,},
Hideki Murakawa ^{2
,}

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

Received: 01 March 2014 Revised: 01 October 2014
35K05, 35K51, 35K57, 80A22, 80A30.

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.
- Reaction-diffusion system,
- precipitation,
- dissolution,
- singular limit analysis,
- fast reaction limit,
- Stefan problem,
- heat equation
Citation: Danielle Hilhorst, Hideki Murakawa. Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium[J]. Networks and Heterogeneous Media, 2014, 9(4): 669-682. doi: 10.3934/nhm.2014.9.669

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Abstract

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.

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
[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
[3]	H. Brézis, Analyse Fonctionnelle, Masson, 1983.
[4]	R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, handbook of numerical analysis, Handb. Numer. Anal., VII (2000), 713-1020.
[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

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