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Fractional time differential equations as a singular limit of the Kobayashi–Warren–Carter system

Part of: Parabolic equations and systems Phase transformations in solids Miscellaneous topics - Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  11 December 2024

Yoshikazu Giga ,
Ayato Kubo ,
Hirotoshi Kuroda ,
Jun Okamoto ,
Koya Sakakibara Open the ORCID record for Koya Sakakibara [Opens in a new window]  and
Masaaki Uesaka
Yoshikazu Giga
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan (labgiga@ms.u-tokyo.ac.jp.)
Ayato Kubo
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido 060-0810, Japan (kubo.ayato.j8@elms.hokudai.ac.jp)
Hirotoshi Kuroda
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido 060-0810, Japan (kuro@math.sci.hokudai.ac.jp)
Jun Okamoto
Affiliation:
Institute for the Advanced Study of Human Biology, Kyoto University, Yoshida-Konoe-Cho, Sakyo-ku, Kyoto 606-8501, Japan (okamoto.jun.8n@kyoto-u.ac.jp)
Koya Sakakibara*
Affiliation:
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa-shi, Ishikawa 920-1192, Japan RIKEN iTHEMS, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan (ksakaki@se.kanazawa-u.ac.jp) (corresponding author)
Masaaki Uesaka
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan Arithmer Inc., ONEST Hongo Square 3F, 1-24-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Present address: DataLabs Inc., 8-6, Nihonbashi Kobunachou, Chuo-ku, Tokyo 103-0024, Japan (masaaki.uesaka@datalabs.jp)
*
*Corresponding author.
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Abstract

This paper is concerned with a singular limit of the Kobayashi–Warren–Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica–Mortola functional in a one-dimensional setting.

Keywords

fractional time derivativegradient flowKobayashi–Warren–Carter systemsingular limit

MSC classification

Primary: 35R11: Fractional partial differential equations
Secondary: 35Q74: PDEs in connection with mechanics of deformable solids 35K20: Initial-boundary value problems for second-order parabolic equations 74N99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 37
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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