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Degenerate equations in a diffusion–precipitation model for clogging porous media

Part of: Parabolic equations and systems Diffusion and convection Representations of solutions

Published online by Cambridge University Press:  18 December 2019

RAPHAEL SCHULZ
RAPHAEL SCHULZ*
Affiliation:
Friedrich–Alexander Universität Erlangen–Nürnberg, Department of Mathematics, Cauerstraße 11, 91058Erlangen, Germany, email: [email protected]
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Abstract

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In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity. Thereby, the evolution of the microstructure is caused by precipitation of the transported substance. We are particularly interested in analysing the model when the equations degenerate due to clogging. Introducing an appropriate weighted function space, we are able to handle the degeneracy and obtain analytical results for the transport equation. Also the decay behaviour of this solution with respect to the porosity is investigated. There a restriction on the decay order is assumed, that is, besides low initial concentration also dense precipitation leads to possible high decay. We obtain nonnegativity and boundedness for the weak solution to the transport equation. Moreover, we study an ordinary differential equation (ODE) describing the change of porosity. Thereby, the control of an appropriate weighted norm of the gradient of the porosity is crucial for the analysis of the transport equation. In order to obtain global in time solutions to the overall coupled system, we apply a fixed point argument. The problem is solved for substantially degenerating hydrodynamic parameters.

Keywords

Evolving porous mediadegenerate equationcloggingweighted spacesdecay behaviour

MSC classification

Primary: 35D30: Weak solutions
Secondary: 35K65: Degenerate parabolic equations 76R50: Diffusion
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 6 , December 2020 , pp. 1050 - 1069
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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