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Strong solvability up to clogging of an effective diffusion–precipitation model in an evolving porous medium

Published online by Cambridge University Press:  14 April 2016

R. SCHULZ ,
N. RAY ,
F. FRANK ,
H. S. MAHATO  and
P. KNABNER
R. SCHULZ
Affiliation:
Department of Mathematics, Friedrich–Alexander University of Erlangen–Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany email: schulzr@math.fau.de, ray@math.fau.de, knabner@math.fau.de
N. RAY
Affiliation:
Department of Mathematics, Friedrich–Alexander University of Erlangen–Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany email: schulzr@math.fau.de, ray@math.fau.de, knabner@math.fau.de
F. FRANK
Affiliation:
Department of Computational and Applied Mathematics, Rice University, 6100 Main Street–MS 134, Houston, TX 77005-1892, USA email: florian.frank@rice.edu
H. S. MAHATO
Affiliation:
Department of Mathematics, TU Dortmund University, Vogelpothsweg 87, 44227 Dortmund, Germany email: harishankar.mahato@tu-dortmund.de
P. KNABNER
Affiliation:
Department of Mathematics, Friedrich–Alexander University of Erlangen–Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany email: schulzr@math.fau.de, ray@math.fau.de, knabner@math.fau.de
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Abstract

In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one strong solution is shown by applying Schauder's fixed point theorem. Additionally, non-negativity, uniqueness, and global existence or existence up to possible closure of some pores, i.e. up to the limit of degenerating coefficients, is guaranteed.

Keywords

periodic homogenizationevolving microstructureeffective constitutive equationsfinite element methodsstrong solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 2 , April 2017 , pp. 179 - 207
Copyright
Copyright © Cambridge University Press 2016 

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