American Institute of Mathematical Sciences

Advanced
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.  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

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.  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

[1]

Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49
[2]

Yan-Yu Chen, Yoshihito Kohsaka, Hirokazu Ninomiya. Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 697-714. doi: 10.3934/dcdsb.2014.19.697
[3]

E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39
[4]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[5]

María del Mar González, Regis Monneau. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1255-1286. doi: 10.3934/dcds.2012.32.1255
[6]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079
[7]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85
[8]

Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks & Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369
[9]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020033
[10]

Piotr B. Mucha. Limit of kinetic term for a Stefan problem. Conference Publications, 2007, 2007 (Special) : 741-750. doi: 10.3934/proc.2007.2007.741
[11]

Rebecca McKay, Theodore Kolokolnikov. Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 191-220. doi: 10.3934/dcdsb.2012.17.191
[12]

José A. Carrillo, Laurent Desvillettes, Klemens Fellner. Fast-reaction limit for the inhomogeneous Aizenman-Bak model. Kinetic & Related Models, 2008, 1 (1) : 127-137. doi: 10.3934/krm.2008.1.127
[13]

Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
[14]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[15]

Sebastién Gaucel, Michel Langlais. Some remarks on a singular reaction-diffusion system arising in predator-prey modeling. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 61-72. doi: 10.3934/dcdsb.2007.8.61
[16]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[17]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085
[18]

Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643
[19]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for the Ibragimov-Shabat equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 661-673. doi: 10.3934/dcdss.2016020
[20]

Wanli Yang, Jie Sun, Su Zhang. Analysis of optimal boundary control for a three-dimensional reaction-diffusion system. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 325-344. doi: 10.3934/naco.2017021

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences