# American Institute of Mathematical Sciences

February  2014, 7(1): 53-62. doi: 10.3934/dcdss.2014.7.53

## Crystalline motion of spiral-shaped polygonal curves with a tip motion

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

Received  March 2012 Revised  August 2012 Published  July 2013

In this paper we propose 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. We give a tip motion and discuss the behavior of the solution curves by crystalline curvature flow with a driving force. We show that the solution curve belongs to a suitable class of spiral-shaped curves and also show a time-global existence of the spiral-shaped solutions.
Citation: Tetsuya Ishiwata. Crystalline motion of spiral-shaped polygonal curves with a tip motion. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 53-62. doi: 10.3934/dcdss.2014.7.53
##### 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,, in, 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., LIV (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 Mathematical Monographs, (1993). Google Scholar [8] 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 [9] 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 [10] 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 [11] 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 [12] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity,, Discrete Contin. Dyn. Syst. Supplement, (2011), 717. Google Scholar [13] Y. Marutani, H. Ninomiya and R. Weidenfeld, Traveling curved fronts of anisotropic curvature flows,, Japan Journal of Industrial and Applied Mathematics, 23 (2006), 83. doi: 10.1007/BF03167500. Google Scholar [14] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry,, in, 52 (1991), 321. Google Scholar [15] T. K. Ushijima and S. Yazaki, Convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature $V=K^{\alpha}$,, SIAM J. Numer. Anal., 37 (2000), 500. doi: 10.1137/S0036142997330135. Google Scholar [16] S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion,, Hokkaido Math. J., 30 (2001), 327. 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,, in, 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., LIV (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 Mathematical Monographs, (1993). Google Scholar [8] 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 [9] 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 [10] 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 [11] 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 [12] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity,, Discrete Contin. Dyn. Syst. Supplement, (2011), 717. Google Scholar [13] Y. Marutani, H. Ninomiya and R. Weidenfeld, Traveling curved fronts of anisotropic curvature flows,, Japan Journal of Industrial and Applied Mathematics, 23 (2006), 83. doi: 10.1007/BF03167500. Google Scholar [14] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry,, in, 52 (1991), 321. Google Scholar [15] T. K. Ushijima and S. Yazaki, Convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature $V=K^{\alpha}$,, SIAM J. Numer. Anal., 37 (2000), 500. doi: 10.1137/S0036142997330135. Google Scholar [16] S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion,, Hokkaido Math. J., 30 (2001), 327. Google Scholar
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