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

Tetsuya Ishiwata

doi:10.3934/dcdss.2011.4.865

Article Contents

2011, Volume 4, Issue 4: 865-873. Doi: 10.3934/dcdss.2011.4.865

This issue Previous Article On a new kind of convexity for solutions of parabolic problems Next Article The degenerate drift-diffusion system with the Sobolev critical exponent

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

Tetsuya Ishiwata^1,

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

Received: September 2009

Revised: November 2009

Published: August 2011

Abstract / Introduction Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 34A34, 39A12, 74N05; Secondary: 53A04, 34A26.

Citation:

Related Papers

Cited by

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.