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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

Tetsuya Ishiwata 1,

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.
Keywords: Spirals, crystalline curvature, polygonal curves, motion by curvature..
Mathematics Subject Classification: Primary: 34A34, 39A12, 74N05; Secondary: 53A04, 34A2.
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), 323-391. 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), 299-358. 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), 211-228. 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, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appli., Gakkōtosho, Tokyo, 13 (2000), 64-79.  Google Scholar

[5]

Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., 54 (1996), 727-737.  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), 1-19. doi: 10.1007/BF03167495.  Google Scholar

[7]

M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford, Clarendon Press, 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), 233-253. 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., Series S, 4 (2011), 865-873. 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), 717-726.  Google Scholar

[11]

T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62. 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, Rio de Janeiro, Pitman Monographs Surveys Pure Appl. Math., 52 (1991), 321-336, Pitman London. Google Scholar

[13]

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

[14]

H. Imai, N. Ishimura and T. K. Ushijima, A crystalline motion of spiral-shaped curves with symmetry, J. Math. Anal. Appl., 240 (1999), 115-127. 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), 797-806. doi: 10.1051/m2an:1999164.  Google Scholar

show all references

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.  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), 299-358. 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), 211-228. 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, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appli., Gakkōtosho, Tokyo, 13 (2000), 64-79.  Google Scholar

[5]

Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., 54 (1996), 727-737.  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), 1-19. doi: 10.1007/BF03167495.  Google Scholar

[7]

M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford, Clarendon Press, 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), 233-253. 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., Series S, 4 (2011), 865-873. 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), 717-726.  Google Scholar

[11]

T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62. 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, Rio de Janeiro, Pitman Monographs Surveys Pure Appl. Math., 52 (1991), 321-336, Pitman London. Google Scholar

[13]

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

[14]

H. Imai, N. Ishimura and T. K. Ushijima, A crystalline motion of spiral-shaped curves with symmetry, J. Math. Anal. Appl., 240 (1999), 115-127. 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), 797-806. doi: 10.1051/m2an:1999164.  Google Scholar

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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
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