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 and 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), 323-391. doi: 10.1007/BF01041068. [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), 299-358. doi: 10.1098/rsta.1951.0006. [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), 211-228. doi: 10.1016/j.jde.2004.02.012. [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 "Free boundary problems: Theory and applications, I" (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appli., 13, Gakkōtosho, Tokyo, (2000) 64-79. [5] Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., LIV (1996), 727-737. [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), 1-19. doi: 10.1007/BF03167495. [7] M. E. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. [8] H. Imai, N. Ishimura and T. K. Ushijima, A crystalline motion of spiral-shaped curves with symmetry, J. Math. Anal. Appl., 240 (1999), 115-127. doi: 10.1006/jmaa.1999.6599. [9] H. Imai, N. Ishimura and T. K. Ushijima, Motion of spirals by crystalline curvature, M2AN Math. Model. Numer. Anal., 33 (1999), 797-806. doi: 10.1051/m2an:1999164. [10] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan Journal of Industrial and Applied Mathematics, 25 (2008), 233-253. doi: 10.1007/BF03167521. [11] T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect, Discrete Contin. Dyn. Syst., Series S, 4 (2011), 865-873. doi: 10.3934/dcdss.2011.4.865. [12] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst. Supplement, (2011), 717-726. [13] Y. Marutani, H. Ninomiya and R. Weidenfeld, Traveling curved fronts of anisotropic curvature flows, Japan Journal of Industrial and Applied Mathematics, 23 (2006), 83-104. doi: 10.1007/BF03167500. [14] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, in "Differential Geometry," Pitman Monographs Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, (1991), 321-336. [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), 500-522. doi: 10.1137/S0036142997330135. [16] S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357.

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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), 323-391. doi: 10.1007/BF01041068. [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), 299-358. doi: 10.1098/rsta.1951.0006. [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), 211-228. doi: 10.1016/j.jde.2004.02.012. [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 "Free boundary problems: Theory and applications, I" (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appli., 13, Gakkōtosho, Tokyo, (2000) 64-79. [5] Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., LIV (1996), 727-737. [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), 1-19. doi: 10.1007/BF03167495. [7] M. E. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. [8] H. Imai, N. Ishimura and T. K. Ushijima, A crystalline motion of spiral-shaped curves with symmetry, J. Math. Anal. Appl., 240 (1999), 115-127. doi: 10.1006/jmaa.1999.6599. [9] H. Imai, N. Ishimura and T. K. Ushijima, Motion of spirals by crystalline curvature, M2AN Math. Model. Numer. Anal., 33 (1999), 797-806. doi: 10.1051/m2an:1999164. [10] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan Journal of Industrial and Applied Mathematics, 25 (2008), 233-253. doi: 10.1007/BF03167521. [11] T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect, Discrete Contin. Dyn. Syst., Series S, 4 (2011), 865-873. doi: 10.3934/dcdss.2011.4.865. [12] T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst. Supplement, (2011), 717-726. [13] Y. Marutani, H. Ninomiya and R. Weidenfeld, Traveling curved fronts of anisotropic curvature flows, Japan Journal of Industrial and Applied Mathematics, 23 (2006), 83-104. doi: 10.1007/BF03167500. [14] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, in "Differential Geometry," Pitman Monographs Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, (1991), 321-336. [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), 500-522. doi: 10.1137/S0036142997330135. [16] S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357.
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