# American Institute of Mathematical Sciences

October  2015, 8(5): 881-888. doi: 10.3934/dcdss.2015.8.881

## On spiral solutions to generalized crystalline motion with a rotating tip motion

 1 Shibaura Institute of Technology, Fukasaku 309, Minuma-ku, Saitama, 337-8570

Received  January 2014 Revised  March 2015 Published  July 2015

In our previous paper we proposed a crystalline motion of spiral-shaped polygonal curves with a tip motion as a simple model of a step motion on a crystal surface under screw dislocation and discussed global existence of spiral solutions to the proposed model. In this paper we extend the previous results for generalized crystalline curvature flow with a suitable tip motion. We show that solution curves never intersect a trajectory of a tip and has no self-intersections. We also show that any facet never disappear during time evolution. Finally we show a time-global existence of the spiral-shaped solutions.
Citation: Tetsuya Ishiwata. On spiral solutions to generalized crystalline motion with a rotating tip motion. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 881-888. doi: 10.3934/dcdss.2015.8.881
##### References:
 [1] S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermal interface,, Arch. Rational Mech. Anal., 108 (1989), 323.  doi: 10.1007/BF01041068.  Google Scholar [2] W. K. Burton, N. Cabrera and F. C. Frank, The Growth of Crystals and the Equilibrium Structure of their Surfaces,, Philos. Trans. Roy. Soc. London. Ser. A., 243 (1951), 299.  doi: 10.1098/rsta.1951.0006.  Google Scholar [3] B. Fiedler, J.-S. Guo and J.-C. Tsai, Multiplicity of rotating spirals under curvature flows with normal tip motion,, J. Differential Equations, 205 (2004), 211.  doi: 10.1016/j.jde.2004.02.012.  Google Scholar [4] 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, 13 (2000), 64.   Google Scholar [5] Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane,, Quart. J. Appl. Math., 54 (1996), 727.   Google Scholar [6] J.-S. Guo, K.-I. Nakamura, T. Ogiwara and J.-C. Tsai, On the Steadily Rotating Spirals,, Japan J. Indust. Appl. Math., 23 (2006), 1.  doi: 10.1007/BF03167495.  Google Scholar [7] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane,, Oxford, (1993).   Google Scholar [8] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena,, Japan Journal of Industrial and Applied Mathematics, 25 (2008), 233.  doi: 10.1007/BF03167521.  Google Scholar [9] T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect,, Discrete Contin. Dyn. Syst., 4 (2011), 865.  doi: 10.3934/dcdss.2011.4.865.  Google Scholar [10] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity,, Discrete Contin. Dyn. Syst. Supplement, 1 (2011), 717.   Google Scholar [11] T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53.  doi: 10.3934/dcdss.2014.7.53.  Google Scholar [12] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry,, Proceedings of the Conference on Differential Geometry, 52 (1991), 321.   Google Scholar [13] S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion,, Hokkaido Math. J., 30 (2001), 327.  doi: 10.14492/hokmj/1350911957.  Google Scholar [14] H. Imai, N. Ishimura and T. K. Ushijima, A crystalline motion of spiral-shaped curves with symmetry,, J. Math. Anal. Appl., 240 (1999), 115.  doi: 10.1006/jmaa.1999.6599.  Google Scholar [15] H. Imai, N. Ishimura and T. K. Ushijima, Motion of spirals by crystalline curvature,, M2AN Math. Model. Numer. Anal., 33 (1999), 797.  doi: 10.1051/m2an:1999164.  Google Scholar

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##### References:
 [1] S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermal interface,, Arch. Rational Mech. Anal., 108 (1989), 323.  doi: 10.1007/BF01041068.  Google Scholar [2] W. K. Burton, N. Cabrera and F. C. Frank, The Growth of Crystals and the Equilibrium Structure of their Surfaces,, Philos. Trans. Roy. Soc. London. Ser. A., 243 (1951), 299.  doi: 10.1098/rsta.1951.0006.  Google Scholar [3] B. Fiedler, J.-S. Guo and J.-C. Tsai, Multiplicity of rotating spirals under curvature flows with normal tip motion,, J. Differential Equations, 205 (2004), 211.  doi: 10.1016/j.jde.2004.02.012.  Google Scholar [4] 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, 13 (2000), 64.   Google Scholar [5] Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane,, Quart. J. Appl. Math., 54 (1996), 727.   Google Scholar [6] J.-S. Guo, K.-I. Nakamura, T. Ogiwara and J.-C. Tsai, On the Steadily Rotating Spirals,, Japan J. Indust. Appl. Math., 23 (2006), 1.  doi: 10.1007/BF03167495.  Google Scholar [7] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane,, Oxford, (1993).   Google Scholar [8] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena,, Japan Journal of Industrial and Applied Mathematics, 25 (2008), 233.  doi: 10.1007/BF03167521.  Google Scholar [9] T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect,, Discrete Contin. Dyn. Syst., 4 (2011), 865.  doi: 10.3934/dcdss.2011.4.865.  Google Scholar [10] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity,, Discrete Contin. Dyn. Syst. Supplement, 1 (2011), 717.   Google Scholar [11] T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53.  doi: 10.3934/dcdss.2014.7.53.  Google Scholar [12] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry,, Proceedings of the Conference on Differential Geometry, 52 (1991), 321.   Google Scholar [13] S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion,, Hokkaido Math. J., 30 (2001), 327.  doi: 10.14492/hokmj/1350911957.  Google Scholar [14] H. Imai, N. Ishimura and T. K. Ushijima, A crystalline motion of spiral-shaped curves with symmetry,, J. Math. Anal. Appl., 240 (1999), 115.  doi: 10.1006/jmaa.1999.6599.  Google Scholar [15] H. Imai, N. Ishimura and T. K. Ushijima, Motion of spirals by crystalline curvature,, M2AN Math. Model. Numer. Anal., 33 (1999), 797.  doi: 10.1051/m2an:1999164.  Google Scholar
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