Volume 55, Number 4, July-August 2021
|Page(s)||1669 - 1697|
|Published online||10 August 2021|
Existence of traveling wave solutions for the Diffusion Poisson Coupled Model: a computer-assisted proof
Centre de mathématiques appliquées, Ecole Polytechnique, route de Saclay, 91128 Palaiseau Cedex, France.
2 Univ. Lille, CNRS, UMR 8524, Inria – Laboratoire Paul Painlevé, F-59000 Lille, France.
3 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraß e 8–10, 1040 Wien, Austria.
* Corresponding author: firstname.lastname@example.org
Accepted: 13 July 2021
The Diffusion Poisson Coupled Model describes the evolution of a dense oxide layer appearing at the surface of carbon steel canisters in contact with a claystone formation. This model is a one dimensional free boundary problem involving drift-diffusion equations on the density of species (electrons, ferric cations and oxygen vacancies), coupled with a Poisson equation on the electrostatic potential and with moving boundary equations, which describe the evolution of the position of each unknown interfaces of the spatial domain. Numerical simulations suggest the existence of traveling wave solutions for this model. These solutions are defined by stationary profiles on a fixed size domain with interfaces moving both at the same velocity. In this paper, we present and apply a computer-assisted method in order to prove the existence of these traveling wave solutions. We also establish a precise and certified description of the solutions.
Mathematics Subject Classification: 35C07 / 35Q92 / 47H10 / 65G20 / 65N35
Key words: Rigorous numerics / corrosion model / traveling wave solutions / spectral methods / fixed-point argument
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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