# 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
##### References:
 [1] F. Almgren and J. E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geom., 42 (1995), 1-22. doi: 10.4310/jdg/1214457030. Google Scholar [2] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-438. doi: 10.1137/0331020. Google Scholar [3] S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure. Ⅱ. Evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391. doi: 10.1007/BF01041068. Google Scholar [4] W. K. Burton, N. Cabrera and F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 243 (1951), 299-358. doi: 10.1098/rsta.1951.0006. Google Scholar [5] R. E. Caflisch, Growth, structure and pattern formation for thin films, J. Sci. Comput., 37 (2008), 3-17. doi: 10.1007/s10915-008-9206-8. Google Scholar [6] A. Chambolle, M. Morini, M. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flow, to appear in J. Amer. Math. Soc., arXiv: 1702.03094.Google Scholar [7] A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), 195-218. doi: 10.4171/IFB/97. Google Scholar [8] A. Chambolle, M. Morini and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow, Comm. Pure Appl. Math., 70 (2017), 1084-1114. doi: 10.1002/cpa.21668. Google Scholar [9] B. Engquist, A.-K. Tornberg and R. Tsai, Discretization of Dirac delta functions in level set methods, J. Comput. Phys., 207 (2005), 28-51. doi: 10.1016/j.jcp.2004.09.018. Google Scholar [10] N. Forcadel, C. Imbert and R. Monneau, Uniqueness and existence of spirals moving by forced mean curvature motion, Interfaces Free Bound., 14 (2012), 365-400. doi: 10.4171/IFB/285. Google Scholar [11] M.-H. Giga and Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal., 159 (2001), 295-333. doi: 10.1007/s002050100154. Google Scholar [12] Y. Giga, Surface Evolution Equations: A Level Set Approach, Monographs in Mathematics, 99, Birkhäuser Verlag, Basel, 2006. Google Scholar [13] Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces, Adv. Differential Equations, 21 (2016), 631-698. Google Scholar [14] Y. Giga and N. Požár, Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to crystalline mean curvature flow, Comm. Pure Appl. Math., 71 (2018), 1461-1491. doi: 10.1002/cpa.21752. Google Scholar [15] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314. doi: 10.4310/jdg/1214441371. Google Scholar [16] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993. [17] H. Imai, N. Ishimura and T. Ushijima, Motion of spirals by crystalline curvature, M2AN Math. Model. Numer. Anal., 33 (1999), 797-806. doi: 10.1051/m2an:1999164. Google Scholar [18] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan J. Indust. Appl. Math., 25 (2008), 233-253. doi: 10.1007/BF03167521. Google Scholar [19] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst., 2011,717–726. doi: 10.3934/proc.2011.2011.717. Google Scholar [20] T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 865-873. doi: 10.3934/dcdss.2011.4.865. Google Scholar [21] T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62. doi: 10.3934/dcdss.2014.7.53. Google Scholar [22] T. Ishiwata, T. K. Ushijima, H. Yagisita and S. Yazaki, Two examples of nonconvex self-similar solution curves for a crystalline curvature flow, Proc. Japan Acad. Ser. A Math. Sci., 80 (2004), 151-154. doi: 10.3792/pjaa.80.151. Google Scholar [23] A. Karma and M. Plapp, Spiral surface growth without desorption, Phys. Rev. Lett., 81 (1998), 4444-4447. doi: 10.1103/PhysRevLett.81.4444. Google Scholar [24] R. Kobayashi, A brief introduction to phase field method, AIP Conf. Proc., 1270 (2010), 282-291. doi: 10.1063/1.3476232. Google Scholar [25] H. Miura and R. Kobayashi, Phase-field modeling of step dynamics on growing crystal surface: Direct integration of growth units to step front, Crystal Growth & Design, 15 (2015), 2165-2175. doi: 10.1021/cg501806d. Google Scholar [26] A. Oberman, S. Osher, R. Takei and R. Tsai, Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation, Commun. Math. Sci., 9 (2011), 637-662. doi: 10.4310/CMS.2011.v9.n3.a1. Google Scholar [27] T. Ohtsuka, Y.-H. Tsai and Y. Giga, A level set approach reflecting sheet structure with single auxiliary function for evolving spirals on crystal surfaces, Journal of Scientific Computing, 62 (2015), 831-874. doi: 10.1007/s10915-014-9877-2. Google Scholar [28] T. Ohtsuka, A level set method for spiral crystal growth, Advances in Mathematical Sciences and Applications, 13 (2003), 225-248. Google Scholar [29] T. Ohtsuka, Minimizing movement approach for spirals evolving by crystalline curvature using level set functions, Oberwolfach Reports, 14 (2017), 314-317. Google Scholar [30] A. G. Shtukenberg, Z. Zhu, Z. An, M. Bhandari, P. Song, B. Kahr and M. D. Ward, Illusory spirals and loops in crystal growth, Proc. Natl. Acad. Sci. USA, 110 (2013), 17195-17198. doi: 10.1073/pnas.1311637110. Google Scholar [31] P. Smereka, Spiral crystal growth, Physica D. Nonlinear Phenomena, 138 (2000), 282-301. doi: 10.1016/S0167-2789(99)00216-X. Google Scholar [32] I. Sunagawa and P. Bennema, Morphology of growth spirals: Theoretical and experimental, in Preparation and Properties of Solid State Materials: Vol. 7, Growth Mechanisms and Silicon Nitride, 1–129.Google Scholar [33] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, in Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, 1991,321–336. Google Scholar [34] S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357. doi: 10.14492/hokmj/1350911957. Google Scholar

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##### References:
 [1] F. Almgren and J. E. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geom., 42 (1995), 1-22. doi: 10.4310/jdg/1214457030. Google Scholar [2] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-438. doi: 10.1137/0331020. Google Scholar [3] S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure. Ⅱ. Evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391. doi: 10.1007/BF01041068. Google Scholar [4] W. K. Burton, N. Cabrera and F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 243 (1951), 299-358. doi: 10.1098/rsta.1951.0006. Google Scholar [5] R. E. Caflisch, Growth, structure and pattern formation for thin films, J. Sci. Comput., 37 (2008), 3-17. doi: 10.1007/s10915-008-9206-8. Google Scholar [6] A. Chambolle, M. Morini, M. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flow, to appear in J. Amer. Math. Soc., arXiv: 1702.03094.Google Scholar [7] A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), 195-218. doi: 10.4171/IFB/97. Google Scholar [8] A. Chambolle, M. Morini and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow, Comm. Pure Appl. Math., 70 (2017), 1084-1114. doi: 10.1002/cpa.21668. Google Scholar [9] B. Engquist, A.-K. Tornberg and R. Tsai, Discretization of Dirac delta functions in level set methods, J. Comput. Phys., 207 (2005), 28-51. doi: 10.1016/j.jcp.2004.09.018. Google Scholar [10] N. Forcadel, C. Imbert and R. Monneau, Uniqueness and existence of spirals moving by forced mean curvature motion, Interfaces Free Bound., 14 (2012), 365-400. doi: 10.4171/IFB/285. Google Scholar [11] M.-H. Giga and Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal., 159 (2001), 295-333. doi: 10.1007/s002050100154. Google Scholar [12] Y. Giga, Surface Evolution Equations: A Level Set Approach, Monographs in Mathematics, 99, Birkhäuser Verlag, Basel, 2006. Google Scholar [13] Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces, Adv. Differential Equations, 21 (2016), 631-698. Google Scholar [14] Y. Giga and N. Požár, Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to crystalline mean curvature flow, Comm. Pure Appl. Math., 71 (2018), 1461-1491. doi: 10.1002/cpa.21752. Google Scholar [15] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314. doi: 10.4310/jdg/1214441371. Google Scholar [16] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993. [17] H. Imai, N. Ishimura and T. Ushijima, Motion of spirals by crystalline curvature, M2AN Math. Model. Numer. Anal., 33 (1999), 797-806. doi: 10.1051/m2an:1999164. Google Scholar [18] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan J. Indust. Appl. Math., 25 (2008), 233-253. doi: 10.1007/BF03167521. Google Scholar [19] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst., 2011,717–726. doi: 10.3934/proc.2011.2011.717. Google Scholar [20] T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 865-873. doi: 10.3934/dcdss.2011.4.865. Google Scholar [21] T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62. doi: 10.3934/dcdss.2014.7.53. Google Scholar [22] T. Ishiwata, T. K. Ushijima, H. Yagisita and S. Yazaki, Two examples of nonconvex self-similar solution curves for a crystalline curvature flow, Proc. Japan Acad. Ser. A Math. Sci., 80 (2004), 151-154. doi: 10.3792/pjaa.80.151. Google Scholar [23] A. Karma and M. Plapp, Spiral surface growth without desorption, Phys. Rev. Lett., 81 (1998), 4444-4447. doi: 10.1103/PhysRevLett.81.4444. Google Scholar [24] R. Kobayashi, A brief introduction to phase field method, AIP Conf. Proc., 1270 (2010), 282-291. doi: 10.1063/1.3476232. Google Scholar [25] H. Miura and R. Kobayashi, Phase-field modeling of step dynamics on growing crystal surface: Direct integration of growth units to step front, Crystal Growth & Design, 15 (2015), 2165-2175. doi: 10.1021/cg501806d. Google Scholar [26] A. Oberman, S. Osher, R. Takei and R. Tsai, Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation, Commun. Math. Sci., 9 (2011), 637-662. doi: 10.4310/CMS.2011.v9.n3.a1. Google Scholar [27] T. Ohtsuka, Y.-H. Tsai and Y. Giga, A level set approach reflecting sheet structure with single auxiliary function for evolving spirals on crystal surfaces, Journal of Scientific Computing, 62 (2015), 831-874. doi: 10.1007/s10915-014-9877-2. Google Scholar [28] T. Ohtsuka, A level set method for spiral crystal growth, Advances in Mathematical Sciences and Applications, 13 (2003), 225-248. Google Scholar [29] T. Ohtsuka, Minimizing movement approach for spirals evolving by crystalline curvature using level set functions, Oberwolfach Reports, 14 (2017), 314-317. Google Scholar [30] A. G. Shtukenberg, Z. Zhu, Z. An, M. Bhandari, P. Song, B. Kahr and M. D. Ward, Illusory spirals and loops in crystal growth, Proc. Natl. Acad. Sci. USA, 110 (2013), 17195-17198. doi: 10.1073/pnas.1311637110. Google Scholar [31] P. Smereka, Spiral crystal growth, Physica D. Nonlinear Phenomena, 138 (2000), 282-301. doi: 10.1016/S0167-2789(99)00216-X. Google Scholar [32] I. Sunagawa and P. Bennema, Morphology of growth spirals: Theoretical and experimental, in Preparation and Properties of Solid State Materials: Vol. 7, Growth Mechanisms and Silicon Nitride, 1–129.Google Scholar [33] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, in Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, 1991,321–336. Google Scholar [34] S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357. doi: 10.14492/hokmj/1350911957. Google Scholar
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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