August  2011, 4(4): 865-873. doi: 10.3934/dcdss.2011.4.865

On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect

1. 

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

Received  September 2009 Revised  November 2009 Published  November 2010

The behavior of polygonal curves with asymptotic lines to crystalline motion with the bulk effect is discussed. We show sufficient conditions for global existence of the solutions and characterize facet-extinction patterns. We also show the eventual monotonicity of shape of the solution curves, that is, the solutions become V-shaped in finite time.
Citation: Tetsuya Ishiwata. On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 865-873. doi: 10.3934/dcdss.2011.4.865
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Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., LIV (1996), 727-737.

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M. E. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane," Oxford, Clarendon Press, 1993.

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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: doi:10.1007/BF03167521.

[5]

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: doi:10.1007/BF03167500.

[6]

S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357.

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: doi:10.1007/BF01041068.

[2]

Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., LIV (1996), 727-737.

[3]

M. E. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane," Oxford, Clarendon Press, 1993.

[4]

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: doi:10.1007/BF03167521.

[5]

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: doi:10.1007/BF03167500.

[6]

S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J., 30 (2001), 327-357.

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