# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021182
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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

Received  March 2021 Revised  September 2021 Early access November 2021

Fund Project: 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

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.

Citation: Mi-Ho Giga, Yoshikazu Giga, Ryo Kuroda, Yusuke Ochiai. Crystalline flow starting from a general polygon. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021182
##### References:
 [1] B. Andrews, Singularities in crystalline curvature flows, Asian J. Math., 6 (2002), 101-121.  doi: 10.4310/AJM.2002.v6.n1.a6.  Google Scholar [2] 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 [3] G. Bellettini, M. Novaga and M. Paolini, Facet-breaking for three-dimensional crystals evolving by mean curvature, Interfaces Free Bound., 1 (1999), 39-55.  doi: 10.4171/IFB/3.  Google Scholar [4] G. Bellettini, M. Novaga and M. Paolini, Characterization of facet breaking for nonsmooth mean curvature flow in the convex case, Interfaces Free Bound., 3 (2001), 415-446.  doi: 10.4171/IFB/47.  Google Scholar [5] D. Campbell, A First Glance at Crystal Motion, Master's thesis, Rutgers University, New Brunswick, NJ, 2002. Google Scholar [6] A. Chambolle, M. Morini, M. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, J. Amer. Math. Soc., 32 (2019), 779-824.  doi: 10.1090/jams/919.  Google Scholar [7] 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 [8] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.   Google Scholar [9] C. Dohmen and Y. Giga, Selfsimilar shrinking curves for anisotropic curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci., 70 (1994), 252-255.   Google Scholar [10] C. Dohmen, Y. Giga and N. Mizoguchi, Existence of selfsimilar shrinking curves for anisotropic curvature flow equations, Calc. Var. Partial Differential Equations, 4 (1996), 103-119.  doi: 10.1007/BF01189949.  Google Scholar [11] C. M. Elliott, A. R. Gardiner and R. Schätzle, Crystalline curvature flow of a graph in a variational setting, Adv. Math. Sci. Appl., 8 (1998), 425-460.   Google Scholar [12] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.   Google Scholar [13] T. Fukui and Y. Giga, Motion of a graph by nonsmooth weighted curvature, In World Congress of Nonlinear Analysts '92, Vol. I–IV (Tampa, FL, 1992), de Gruyter, Berlin, (1996), 47–56.  Google Scholar [14] M. E. Gage, Evolving plane curves by curvature in relative geometries, Duke Math. J., 72 (1993), 441-466.  doi: 10.1215/S0012-7094-93-07216-X.  Google Scholar [15] M. E. Gage and Y. Li, Evolving plane curves by curvature in relative geometries. II, Duke Math. J., 75 (1994), 79-98.  doi: 10.1215/S0012-7094-94-07503-0.  Google Scholar [16] R. Gérard and H. Tahara, Singular Nonlinear Partial Differential Equations, Aspects of Mathematics Friedr. Vieweg & Sohn, Braunschweig, 1996. doi: 10.1007/978-3-322-80284-2.  Google Scholar [17] M.-H. Giga and Y. Giga, Consistency in evolutions by crystalline curvature, Free Boundary Problems, Theory and Applications (Zakopane, 1995), Pitman Res. Notes Math. Ser., Longman, Harlow, 363 (1996), 186–202.  Google Scholar [18] M.-H. Giga and Y. Giga, Geometric evolution by nonsmooth interfacial energy, Nonlinear Analysis and Applications (Warsaw, 1994), GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 7 (1996), 125–140.  Google Scholar [19] M.-H. Giga and Y. Giga, A subdifferential interpretation of crystalline motion under nonuniform driving force, Discrete Contin. Dynam. Systems, 1 (1998), 276-287.   Google Scholar [20] M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal., 141 (1998), 117-198.  doi: 10.1007/s002050050075.  Google Scholar [21] M.-H. Giga and Y. Giga, Stability for evolving graphs by nonlocal weighted curvature, Comm. Partial Differential Equations, 24 (1999), 109-184.  doi: 10.1080/03605309908821419.  Google Scholar [22] M.-H. Giga and Y. Giga, Crystalline and level set flow – convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane, Free Boundary Problems: Theory and Applications, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 13 (2000), 64–79.  Google Scholar [23] 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 [24] M.-H. Giga and Y. Giga, On the role of kinetic and interfacial anisotropy in the crystal growth theory, Interfaces Free Bound., 15 (2013), 429-450.  doi: 10.4171/IFB/309.  Google Scholar [25] M.-H. Giga, Y. Giga and H. Hontani, Self-similar expanding solutions in a sector for a crystalline flow, SIAM J. Math. Anal., 37 (2005), 1207-1226.  doi: 10.1137/040614372.  Google Scholar [26] Y. Giga, Motion of a graph by convexified energy, Hokkaido Math. J., 23 (1994), 185-212.  doi: 10.14492/hokmj/1381412492.  Google Scholar [27] Y. Giga, Surface Evolution Equations. A Level Set Approach, Monographs in Mathematics, 99, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7391-1.  Google Scholar [28] Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. Appl. Math., 54 (1996), 727-737.  doi: 10.1090/qam/1417236.  Google Scholar [29] Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces, Adv. Differential Equations, 21 (2016), 631-698.   Google Scholar [30] 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 [31] Y. Giga and N. Požár, Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term, SN Partial Differ. Equ. Appl., 1 (2020), Article number: 39. doi: 10.1007/s42985-020-00040-0.  Google Scholar [32] P. M. Girão, Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature, SIAM J. Numer. Anal., 32 (1995), 886-899.  doi: 10.1137/0732041.  Google Scholar [33] P. M. Girão and R. V. Kohn, Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature, Numer. Math., 67 (1994), 41-70.  doi: 10.1007/s002110050017.  Google Scholar [34] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.   Google Scholar [35] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993.   Google Scholar [36] M. E. Gurtin, H. M. Soner and P. E. Souganidis, Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles, J. Differential Equations, 119 (1995), 54-108.  doi: 10.1006/jdeq.1995.1084.  Google Scholar [37] K. Ishii and H. M. Soner, Regularity and convergence of crystalline motion, SIAM J. Math. Anal., 30 (1999), 19-37.  doi: 10.1137/S0036141097317347.  Google Scholar [38] T. Ishiwata, Crystalline Undou ni Tsuite: Heimenjou No Takakukei No Undou No Kaiseki, (Japanese) [On crystalline motion: Analysis on motion of a polygon in the plane], Lecture Series in Mathematics GP-TML06, Graduate School of Science, Tohoku University, 2008. Google Scholar [39] 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 [40] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8$^{th}$ AIMS Conference. 1 (2011), 717–726.  Google Scholar [41] 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 [42] 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 [43] T. Ishiwata and T. Ohtsuka, Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5261-5295.  doi: 10.3934/dcdsb.2019058.  Google Scholar [44] T. Ishiwata and T. Ohtsuka, Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 893-907.  doi: 10.3934/dcdss.2020390.  Google Scholar [45] 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.   Google Scholar [46] R. Kuroda, Facet-Creation Between Two Facets Moved by Crystalline Flow or Similar Equations, Bachelor's thesis, The University of Tokyo, Tokyo, 2019. Google Scholar [47] P. B. Mucha, Regular solutions to a monodimensional model with discontinuous elliptic operator, Interfaces Free Bound., 14 (2012), 145-152.  doi: 10.4171/IFB/276.  Google Scholar [48] P. B. Mucha and P. Rybka, A note on a model system with sudden directional diffusion, J. Stat. Phys., 146 (2012), 975-988.  doi: 10.1007/s10955-012-0446-5.  Google Scholar [49] P. B. Mucha and P. Rybka, Well posedness of sudden directional diffusion equations, Math. Methods Appl. Sci., 36 (2013), 2359-2370.  doi: 10.1002/mma.2759.  Google Scholar [50] 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 [51] Y. Ochiai, Facet-Creation Between Two Facets Moved by Crystalline Curvature, Master's thesis, The University of Tokyo, Tokyo, 2009. Google Scholar [52] A. Stancu, Uniqueness of self-similar solutions for a crystalline flow, Indiana Univ. Math. J., 45 (1996), 1157-1174.  doi: 10.1512/iumj.1996.45.1159.  Google Scholar [53] A. Stancu, Asymptotic behavior of solutions to a crystalline flow, Hokkaido Math. J., 27 (1998), 303-320.  doi: 10.14492/hokmj/1351001287.  Google Scholar [54] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math. Longman Sci. Tech. Harlow, 52 (1991), 321–336.  Google Scholar [55] J. E. Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater., 40 (1992), 1475-1485.  doi: 10.1016/0956-7151(92)90091-R.  Google Scholar [56] J. E. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points, Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Part 1, Amer. Math. Soc., Providence, RI, 54 (1993), 417–438. doi: 10.1090/pspum/054.1/1216599.  Google Scholar [57] S. Yazaki, On an area-preserving crystalline motion, Calc. Var. Partial Differential Equations, 14 (2002), 85-105.  doi: 10.1007/s005260100094.  Google Scholar [58] S. Yazaki, Motion of nonadmissible convex polygons by crystalline curvature, Publ. Res. Inst. Math. Sci., 43 (2007), 155-170.  doi: 10.2977/prims/1199403812.  Google Scholar

show all references

##### References:
 [1] B. Andrews, Singularities in crystalline curvature flows, Asian J. Math., 6 (2002), 101-121.  doi: 10.4310/AJM.2002.v6.n1.a6.  Google Scholar [2] 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 [3] G. Bellettini, M. Novaga and M. Paolini, Facet-breaking for three-dimensional crystals evolving by mean curvature, Interfaces Free Bound., 1 (1999), 39-55.  doi: 10.4171/IFB/3.  Google Scholar [4] G. Bellettini, M. Novaga and M. Paolini, Characterization of facet breaking for nonsmooth mean curvature flow in the convex case, Interfaces Free Bound., 3 (2001), 415-446.  doi: 10.4171/IFB/47.  Google Scholar [5] D. Campbell, A First Glance at Crystal Motion, Master's thesis, Rutgers University, New Brunswick, NJ, 2002. Google Scholar [6] A. Chambolle, M. Morini, M. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, J. Amer. Math. Soc., 32 (2019), 779-824.  doi: 10.1090/jams/919.  Google Scholar [7] 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 [8] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.   Google Scholar [9] C. Dohmen and Y. Giga, Selfsimilar shrinking curves for anisotropic curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci., 70 (1994), 252-255.   Google Scholar [10] C. Dohmen, Y. Giga and N. Mizoguchi, Existence of selfsimilar shrinking curves for anisotropic curvature flow equations, Calc. Var. Partial Differential Equations, 4 (1996), 103-119.  doi: 10.1007/BF01189949.  Google Scholar [11] C. M. Elliott, A. R. Gardiner and R. Schätzle, Crystalline curvature flow of a graph in a variational setting, Adv. Math. Sci. Appl., 8 (1998), 425-460.   Google Scholar [12] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.   Google Scholar [13] T. Fukui and Y. Giga, Motion of a graph by nonsmooth weighted curvature, In World Congress of Nonlinear Analysts '92, Vol. I–IV (Tampa, FL, 1992), de Gruyter, Berlin, (1996), 47–56.  Google Scholar [14] M. E. Gage, Evolving plane curves by curvature in relative geometries, Duke Math. J., 72 (1993), 441-466.  doi: 10.1215/S0012-7094-93-07216-X.  Google Scholar [15] M. E. Gage and Y. Li, Evolving plane curves by curvature in relative geometries. II, Duke Math. J., 75 (1994), 79-98.  doi: 10.1215/S0012-7094-94-07503-0.  Google Scholar [16] R. Gérard and H. Tahara, Singular Nonlinear Partial Differential Equations, Aspects of Mathematics Friedr. Vieweg & Sohn, Braunschweig, 1996. doi: 10.1007/978-3-322-80284-2.  Google Scholar [17] M.-H. Giga and Y. Giga, Consistency in evolutions by crystalline curvature, Free Boundary Problems, Theory and Applications (Zakopane, 1995), Pitman Res. Notes Math. Ser., Longman, Harlow, 363 (1996), 186–202.  Google Scholar [18] M.-H. Giga and Y. Giga, Geometric evolution by nonsmooth interfacial energy, Nonlinear Analysis and Applications (Warsaw, 1994), GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 7 (1996), 125–140.  Google Scholar [19] M.-H. Giga and Y. Giga, A subdifferential interpretation of crystalline motion under nonuniform driving force, Discrete Contin. Dynam. Systems, 1 (1998), 276-287.   Google Scholar [20] M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal., 141 (1998), 117-198.  doi: 10.1007/s002050050075.  Google Scholar [21] M.-H. Giga and Y. Giga, Stability for evolving graphs by nonlocal weighted curvature, Comm. Partial Differential Equations, 24 (1999), 109-184.  doi: 10.1080/03605309908821419.  Google Scholar [22] M.-H. Giga and Y. Giga, Crystalline and level set flow – convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane, Free Boundary Problems: Theory and Applications, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 13 (2000), 64–79.  Google Scholar [23] 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 [24] M.-H. Giga and Y. Giga, On the role of kinetic and interfacial anisotropy in the crystal growth theory, Interfaces Free Bound., 15 (2013), 429-450.  doi: 10.4171/IFB/309.  Google Scholar [25] M.-H. Giga, Y. Giga and H. Hontani, Self-similar expanding solutions in a sector for a crystalline flow, SIAM J. Math. Anal., 37 (2005), 1207-1226.  doi: 10.1137/040614372.  Google Scholar [26] Y. Giga, Motion of a graph by convexified energy, Hokkaido Math. J., 23 (1994), 185-212.  doi: 10.14492/hokmj/1381412492.  Google Scholar [27] Y. Giga, Surface Evolution Equations. A Level Set Approach, Monographs in Mathematics, 99, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7391-1.  Google Scholar [28] Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. Appl. Math., 54 (1996), 727-737.  doi: 10.1090/qam/1417236.  Google Scholar [29] Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces, Adv. Differential Equations, 21 (2016), 631-698.   Google Scholar [30] 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 [31] Y. Giga and N. Požár, Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term, SN Partial Differ. Equ. Appl., 1 (2020), Article number: 39. doi: 10.1007/s42985-020-00040-0.  Google Scholar [32] P. M. Girão, Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature, SIAM J. Numer. Anal., 32 (1995), 886-899.  doi: 10.1137/0732041.  Google Scholar [33] P. M. Girão and R. V. Kohn, Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature, Numer. Math., 67 (1994), 41-70.  doi: 10.1007/s002110050017.  Google Scholar [34] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.   Google Scholar [35] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993.   Google Scholar [36] M. E. Gurtin, H. M. Soner and P. E. Souganidis, Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles, J. Differential Equations, 119 (1995), 54-108.  doi: 10.1006/jdeq.1995.1084.  Google Scholar [37] K. Ishii and H. M. Soner, Regularity and convergence of crystalline motion, SIAM J. Math. Anal., 30 (1999), 19-37.  doi: 10.1137/S0036141097317347.  Google Scholar [38] T. Ishiwata, Crystalline Undou ni Tsuite: Heimenjou No Takakukei No Undou No Kaiseki, (Japanese) [On crystalline motion: Analysis on motion of a polygon in the plane], Lecture Series in Mathematics GP-TML06, Graduate School of Science, Tohoku University, 2008. Google Scholar [39] 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 [40] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8$^{th}$ AIMS Conference. 1 (2011), 717–726.  Google Scholar [41] 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 [42] 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 [43] T. Ishiwata and T. Ohtsuka, Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5261-5295.  doi: 10.3934/dcdsb.2019058.  Google Scholar [44] T. Ishiwata and T. Ohtsuka, Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 893-907.  doi: 10.3934/dcdss.2020390.  Google Scholar [45] 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.   Google Scholar [46] R. Kuroda, Facet-Creation Between Two Facets Moved by Crystalline Flow or Similar Equations, Bachelor's thesis, The University of Tokyo, Tokyo, 2019. Google Scholar [47] P. B. Mucha, Regular solutions to a monodimensional model with discontinuous elliptic operator, Interfaces Free Bound., 14 (2012), 145-152.  doi: 10.4171/IFB/276.  Google Scholar [48] P. B. Mucha and P. Rybka, A note on a model system with sudden directional diffusion, J. Stat. Phys., 146 (2012), 975-988.  doi: 10.1007/s10955-012-0446-5.  Google Scholar [49] P. B. Mucha and P. Rybka, Well posedness of sudden directional diffusion equations, Math. Methods Appl. Sci., 36 (2013), 2359-2370.  doi: 10.1002/mma.2759.  Google Scholar [50] 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 [51] Y. Ochiai, Facet-Creation Between Two Facets Moved by Crystalline Curvature, Master's thesis, The University of Tokyo, Tokyo, 2009. Google Scholar [52] A. Stancu, Uniqueness of self-similar solutions for a crystalline flow, Indiana Univ. Math. J., 45 (1996), 1157-1174.  doi: 10.1512/iumj.1996.45.1159.  Google Scholar [53] A. Stancu, Asymptotic behavior of solutions to a crystalline flow, Hokkaido Math. J., 27 (1998), 303-320.  doi: 10.14492/hokmj/1351001287.  Google Scholar [54] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math. Longman Sci. Tech. Harlow, 52 (1991), 321–336.  Google Scholar [55] J. E. Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater., 40 (1992), 1475-1485.  doi: 10.1016/0956-7151(92)90091-R.  Google Scholar [56] J. E. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points, Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Part 1, Amer. Math. Soc., Providence, RI, 54 (1993), 417–438. doi: 10.1090/pspum/054.1/1216599.  Google Scholar [57] S. Yazaki, On an area-preserving crystalline motion, Calc. Var. Partial Differential Equations, 14 (2002), 85-105.  doi: 10.1007/s005260100094.  Google Scholar [58] S. Yazaki, Motion of nonadmissible convex polygons by crystalline curvature, Publ. Res. Inst. Math. Sci., 43 (2007), 155-170.  doi: 10.2977/prims/1199403812.  Google Scholar
Each arrow indicates the positive direction
A given Wulff shape and initial condition
Self-similar solution
Wulff shape and newly created facets
$i$-th, $(i-1)$-th and $(i+1)$-th facet
Distance function $d_i^s(t)$
New self-similar solution
Wulff shape
Compare with a self-similar solution
An example of the numerical calculation
Another example of the numerical calculation (enlarged)
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