# American Institute of Mathematical Sciences

## Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip

 1 Department of Mathematical Sciences, Shibaura Institute of Technology, Fukasaku 309, Minuma-ku, Saitama 337-8570, Japan 2 Division of Pure and Applied Science, Faculty of Science and Technology, Gunma University, Aramaki-machi 4-2, Maebashi, 371-8510 Gunma, Japan

* Corresponding author: Takeshi Ohtsuka

Received  November 2017 Revised  September 2018 Published  April 2019

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.

Citation: Tetsuya Ishiwata, Takeshi Ohtsuka. Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019058
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Evolution of polygonal spiral. In the above figures, $L_j$ and $y_j$ denote the facets and vertices of the polygonal spiral, respectively
Notations to the Wulff shape $\mathcal{W}_\gamma$
Example of positive convex (left) and positive concave(right) spiral for hexagonal $\mathcal{W}_\gamma$
Evolution of polygonal curves. The angle of the corners with a round curve is $\theta_{j+1} - \theta_j$
Examples of self-intersection between $\Lambda_i (t)$ $(solid line) and$ \Lambda_j (t) $(dashed line). Note that dots means$ y_i (t) $or$ y_j (t) $, which is a vertex belongs to$ \Lambda_i (t) $or$ \Lambda_j (t) $, respectively The case of facet-vertex intersection with$ \Lambda_j (\bar{t}) \subset \mathcal{O}_i (\bar{t}) $(we omit the notation of$ \bar{t} $in the figure). In the situation of the figure (b), the origin should be on the gray regions including the boundary lines by Corollary 3.4. Thus, the kind of touch like as (b) never occur The case of facet-vertex intersection with$ \Lambda_j \subset \mathcal{I}_i (\bar{t}) $. The facet$ \Lambda_{j+1} $never can be located as the dashed line in (b) Location of$ \Lambda_i (\bar{t}) $,$ \Lambda_{i+1} (\bar{t}) $,$ \Lambda_j (\bar{t}) $and$ \Lambda_{j+1} (\bar{t}) $under the vertex-vertex intersection between$ \Lambda_i (\bar{t}) $and$ \Lambda_j (\bar{t}) $. The above figures illustrate the case when (a)$ \varphi_i < \varphi_j < \varphi_i + \pi $, (b)$ \varphi_i + \pi < \varphi_j < \varphi_{i+1} + \pi $, and (c)$ \varphi_{i+1} + \pi < \varphi_j < \varphi_i + 2 \pi $. The gray regions are where$ \Lambda_{j+1} (\bar{t}) $can be located, and gray shaded regions are where the cross type intersection appears between$ \Lambda_i (\bar{t}) \cup \Lambda_{i+1} (\bar{t}) $and$ \Lambda_j (\bar{t}) \cup \Lambda_{j+1} (\bar{t}) $although$ \Lambda_{j+1} (\bar{t}) $seems to be located from the angle condition The case of facet-facet intersection with$ \Lambda_i (\bar{t}) \not\subset \Lambda_j (\bar{t}) $and$ \Lambda_i (\bar{t}) \not\supset \Lambda_j (\bar{t}) $Evolution of a polygonal spiral with a triangle Wulff shape. The figures illustrate the shape of a spiral at$ t = 0.0, 0.1, 0.5, 1.0 $from top left to bottom right Comparison of the profiles of spirals at$ t = 1 $with respect to different anisotropic mobilities under the same$ \mathcal{W}_\gamma $Comparison of profiles between our ODE system(top) and level set method(bottom) for a square spiral at$ t = 0 $(left),$ t = 1 $(center) and$ t = 2 $(right) Comparison of profiles between our ODE system(top) and level set method(bottom) for a diagonal spiral at$ t = 0 $(left),$ t = 1 $(center) and$ t = 2 $(right) Examination of the orientation of$ L_{k+1} $with hexagonal$ \mathcal{W}_\gamma $when$ d_k (T_{k+1}) = \ell_{k}/U $and$ d_{k-1} (T_{k+1}) < \ell_{k-1} / U $The situation when (39) does not hold in Case (i) The situation case(a) with$ \varphi_{j+1} \in (\varphi_{i+1} , \varphi_i + \pi) \$
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