Crystalline flow starting from a general polygon

Mi-Ho Giga; Yoshikazu Giga; Ryo Kuroda; Yusuke Ochiai

doi:10.3934/dcds.2021182

Article Contents

2022, Volume 42, Issue 4: 2027-2051. Doi: 10.3934/dcds.2021182 open access

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Crystalline flow starting from a general polygon

1.
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
2.
Department of Indian Philosophy and Buddhist Studies, Humanities, Faculty of Letters, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
3.
Airitech Co., Ltd., Masonic 39MT Building, 2-4-5 Azabudai, Minato-ku, Tokyo 106-0041, Japan

^* Corresponding author: Yoshikazu Giga
^* Corresponding author: Yoshikazu Giga

Received: February 28, 2021

Revised: August 31, 2021

Early access: November 2021

Published: April 2022

The work of the second author was partly supported by the Japan Society for the Promotion of Science (JSPS) through the grants KAKENHI No. 19H00639, No. 18H05323, No. 17H01091 and by Arithmer Inc. through collaborative grant

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Abstract

This paper solves a singular initial value problem for a system of ordinary differential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.

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Mathematics Subject Classification: Primary: 35K67; Secondary: 34A12, 53E10, 35C06, 74E15.

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Figure 1. Each arrow indicates the positive direction

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Figure 2. A given Wulff shape and initial condition

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Figure 3. Self-similar solution

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Figure 4. Wulff shape and newly created facets

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Figure 5. $ i $-th, $ (i-1) $-th and $ (i+1) $-th facet

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Figure 6. Distance function $ d_i^s(t) $

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Figure 7. New self-similar solution

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Figure 8. Wulff shape

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Figure 9. Compare with a self-similar solution

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Figure 10. An example of the numerical calculation

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Figure 11. Another example of the numerical calculation (enlarged)

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