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Convergence to a self-similar solution for a one-phase Stefan problem arising in corrosion theory

Part of: Representations of solutions Partial differential equations Miscellaneous topics - Partial differential equations Thermodynamics and heat transfer

Published online by Cambridge University Press:  09 August 2022

M. BOUGUEZZI ,
D. HILHORST Open the ORCID record for D. HILHORST [Opens in a new window] ,
Y. MIYAMOTO  and
J.-F. SCHEID
M. BOUGUEZZI
Affiliation:
Université Paris-Saclay, CEA, Service de la Corrosion et du Comportement des Matériaux dans leur Environnement, Gif-sur-Yvette 91191, France email: Mariembougue22i@gmail.com CNRS and Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Orsay 91405, France email: Danielle.Hilhorst@universite-paris-saclay.fr
D. HILHORST
Affiliation:
CNRS and Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Orsay 91405, France email: Danielle.Hilhorst@universite-paris-saclay.fr
Y. MIYAMOTO
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914 Japan email: miyamoto@ms.u-tokyo.ac.jp
J.-F. SCHEID
Affiliation:
Université de Lorraine, CNRS, Inria, IECL, Nancy F-54000, France email: jean-francois.scheid@univ-lorraine.fr
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Abstract

Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$. We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions.

Keywords

Stefan problemfree-boundary problemself-similar solutionupper and lower solutionslarge-time behaviourcorrosion theory

MSC classification

Primary: 35R35: Free boundary problems 35C06: Self-similar solutions 35B40: Asymptotic behavior of solutions
Secondary: 80A22: Stefan problems, phase changes, etc.
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Issue 4 , August 2023 , pp. 701 - 737
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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