Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T22:25:23.813Z Has data issue: false hasContentIssue false
  • English
  • Français

An existence result for a system of coupled semilinear diffusion-reaction equations with flux boundary conditions

Published online by Cambridge University Press:  10 December 2014

HARI SHANKAR MAHATO  and
MICHAEL BÖHM
HARI SHANKAR MAHATO
Affiliation:
Chair of Applied Mathematics 1, University of Erlangen-Nürnberg, 91058 Erlangen, Germany email: [email protected]
MICHAEL BÖHM
Affiliation:
Center of Industrial Mathematics, University of Bremen, 28359 Bremen, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical species present in a porous medium. Inflow–outflow boundary conditions are considered at the outer boundary and the reactions amongst the species are assumed to be reversible which yield highly nonlinear reaction rate terms. The dissolution of immobile species takes place on the surfaces of the solid parts. Modelling of these processes leads to a system of coupled semilinear partial differential equations together with a system of ordinary differential equations (ODEs) with multi-valued right-hand sides. We prove the global existence of a unique positive weak solution of this model using a regularization technique, Schaefer's fixed point theorem and Lyapunov type arguments.

Keywords

Global solutionsemilinear parabolic equationreversible reactionsLyapunov functionalsmaximal regularity
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 2 , April 2015 , pp. 121 - 142
Copyright
Copyright © Cambridge University Press 2014 

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

[1]Clement, P. & Li, S. (1994) Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Appl. Math. Sci. 3, 17–32.Google Scholar
[2]Duijn, C. J. & Knabner, P. (1997) Travelling wave behaviour of crystal dissolution in porous media flow. Eur. J. Appl. Math. 8, 49–72.Google Scholar
[3]Evans, L. C. (1998) Partial Differential Equations, volume 19. AMS Publication, Providence, Rhode Island.Google Scholar
[4]Fatima, T., Arab, N., Zemskov, E. P. & Muntean, A. (2011) Homogenization of a reaction-diffusion system modeling sulfate corrosion of concrete in locally periodic perforated domains. J. Eng. Math. 69, 261–276.Google Scholar
[5]Hoffmann, J. (2010) Reactive Transport and Mineral Dissolution/Precipitation in Porous Media: Efficient Solution Algorithms, Benchmark Computations and Existence of Global Solutions. Ph.D. Thesis, University of Erlangen-Nürnberg.Google Scholar
[6]Knabner, P. (1986) A free boundary problem arising from the leaching of saline soils. SIAM J. Math. Anal. 17 (3), 610–625.Google Scholar
[7]Knabner, P. and van Duijn, C. J. (1996) Crystal dissolution in porous media flow. J. Appl. Math. Mech. 76, 329–332.Google Scholar
[8]Knabner, P., van Duijn, C. J. & Hengst, S. (1995) An analysis of crystal dissolution fronts in flows through porous media part 1: Compatible boundary conditions. Adv. Water Resour. 18, 171–185.Google Scholar
[9]Kräutle, S. (2008) General multispecies reactive transport problems in porous media. Habilitation Thesis, University of Erlangen-Nürnberg, Germany, http://www1.am.uni-erlangen.de/~kraeutle/habil.pdf.Google Scholar
[10]Kräutle, S. (2011) Existence of global solutions of multicomponent reactive transport problems with mass action kinetics in porous media. J. Appl. Anal. Comput. 1, 497–515.Google Scholar
[11]Kumar, K., Neuss-Radu, M. & Pop, I. S. (2013) Homogenization of a pore scale model for precipitation and dissolution in porous media. CASA Report 13-32, TU Eindhoven.Google Scholar
[12]Mahato, H. S. (2013) Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an H1,p Setting. Doctoral Thesis, University of Bremen, Germany, 2013. http://elib.suub.uni-bremen.de/edocs/00103256-1.pdf.Google Scholar
[13]Mahato, H. S. and Böhm, M. (2013a) Global existence and uniqueness for a system of nonlinear multi-species diffusion-reaction equations in an H 1,p setting. J. Appl. Comput. 3 (4), 357–376.Google Scholar
[14]Mahato, H. S. & Böhm, B. (2013b) Homogenization of a system of semilinear diffusion-reaction equations in an H 1,p setting. Electron. J. Differ. Eqns 2013 (2013)(210), 1–22.Google Scholar
[15]Muntean, A. (2006) A Moving Boundary Problem: Modeling, Analysis and Simulation of Concrete Carbonation. Doctoral Thesis, University of Bremen, Germany.Google Scholar
[16]Peter, M. A. (2006) Mathematical Modelling and Homogenization of Coupled Reaction-Diffusion Systems Taking into Account an Evolution of the Microstructure. Doctoral Thesis, University of Bremen, Germany.Google Scholar
[17]Prüss, J. and Schnaubelt, R. (2001) Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time. J. Math. Anal. Appl. 256, 405–430.Google Scholar
[18]Dintelmann, R. H. and Rehberg, J. (2009) Maximal parabolic regularity for divergence operators including mixed boundary conditions. J. Differ. Eqns 247, 1354–1396.CrossRefGoogle Scholar
[19]Roubíček, T. (2005) Nonlinear Partial Differential Equations with Applications, Birkhäuser Publications, Basel-Boston-Berlin.Google Scholar
[20]Showalter, R. E. (1997) Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, volume 49. American Mathematical Society, Oregon State University, Corvallis, Oregon.Google Scholar
[21]van Duijn, C. J. & Pop, I. S. (2004) Crystals dissolution and precipitation in porous media: Pore scale analysis. J. für die Reine Angew. Math. 577, 171–211.Google Scholar
[22]van Noorden, T. L., Pop, I. S. & Röger, M. (2007) Crystal dissolution and precipitation in porous media: L1-contraction and uniqueness. Discrete Continuous Dyn. Syst. 1013–1020.Google Scholar
[23]van Noorden, T. L. (2009) Crystal precipitation and dissolution in a porous medium: Effective equations and numerical experiments. Multiscale Model. Simul. 7 (3), 1220–1236.Google Scholar
[24]van Noorden, T. L. (2009) Crystal precipitation and dissolution in a thin strip. Eur. J. Appl. Math. 20 (1), 69–91.CrossRefGoogle Scholar
[25]van Noorden, T. L. & Pop., I. S. (1970) A Stefan problem modelling dissolution and precipitation in porous media. IMA J. Appl. Math. 73 (2), 393–411.Google Scholar
[26]Yosida, K. (1970) Functional Analysis, Springer Verlag, Berlin.Google Scholar