TY - JOUR
T1 - Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip
AU - Ishiwata, Tetsuya
AU - Ohtsuka, Takeshi
N1 - Funding Information:
The authors are grateful to the anonymous referees for their valuable comments and suggestions to improve presentation of this paper. The first author is partly supported by JSPS KAKENHI Grant Number 15H03632 and 16H03953. The second author is partly supported by JSPS Grant Kiban(C) 26400158.
Funding Information:
Acknowledgments. The authors are grateful to the anonymous referees for their valuable comments and suggestions to improve presentation of this paper. The first author is partly supported by JSPS KAKENHI Grant Number 15H03632
Publisher Copyright:
© 2019 American Institute of Mathematical Sciences. All rights reserved.
PY - 2019/10
Y1 - 2019/10
N2 - We present a new ODE approach for an evolving polygonal spiral by the crystalline eikonal-curvature flow with a fixed center. In this approach, we introduce a mechanism of new facet generation at the center of the growing spiral, which is based on the theory of two-dimensional nucleation. We prove the existence, uniqueness and intersection free of solution to our formulation globally-in-time. In the proof of the existence we also prove that new facets are generated repeatedly in time. The comparison result of the normal velocity between inner and outer facets with the same normal direction leads intersection-free result. The normal velocities are positive after the next new facet is generated, so that the center is always behind of the moving facets.
AB - We present a new ODE approach for an evolving polygonal spiral by the crystalline eikonal-curvature flow with a fixed center. In this approach, we introduce a mechanism of new facet generation at the center of the growing spiral, which is based on the theory of two-dimensional nucleation. We prove the existence, uniqueness and intersection free of solution to our formulation globally-in-time. In the proof of the existence we also prove that new facets are generated repeatedly in time. The comparison result of the normal velocity between inner and outer facets with the same normal direction leads intersection-free result. The normal velocities are positive after the next new facet is generated, so that the center is always behind of the moving facets.
KW - Crystalline curvature flow
KW - Evolution of convex polygonal spiral
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U2 - 10.3934/dcdsb.2019058
DO - 10.3934/dcdsb.2019058
M3 - Article
AN - SCOPUS:85072586230
SN - 1531-3492
VL - 24
SP - 5261
EP - 5295
JO - Discrete and Continuous Dynamical Systems - Series B
JF - Discrete and Continuous Dynamical Systems - Series B
IS - 10
ER -