American Institute of Mathematical Sciences

Advanced
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

Tetsuya Ishiwata 1,

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.
Keywords: Eventual monotonicity., Crystalline curvature, Crystalline motion.
Mathematics Subject Classification: Primary: 34A34, 39A12, 74N05; Secondary: 53A04, 34A2.
Citation: Tetsuya Ishiwata. On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 865-873. doi: 10.3934/dcdss.2011.4.865
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: doi:10.1007/BF01041068.  Google Scholar

[2]

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

[3]

M. E. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane,", Oxford, (1993).   Google Scholar

[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.  doi: doi:10.1007/BF03167521.  Google Scholar

[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.  doi: doi:10.1007/BF03167500.  Google Scholar

[6]

S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion,, Hokkaido Math. J., 30 (2001), 327.   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.  doi: doi:10.1007/BF01041068.  Google Scholar

[2]

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

[3]

M. E. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane,", Oxford, (1993).   Google Scholar

[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.  doi: doi:10.1007/BF03167521.  Google Scholar

[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.  doi: doi:10.1007/BF03167500.  Google Scholar

[6]

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

[1]

Tetsuya Ishiwata. Motion of polygonal curved fronts by crystalline motion: v-shaped solutions and eventual monotonicity. Conference Publications, 2011, 2011 (Special) : 717-726. doi: 10.3934/proc.2011.2011.717
[2]

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
[3]

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
[4]

Mi-Ho Giga, Yoshikazu Giga. A subdifferential interpretation of crystalline motion under nonuniform driving force. Conference Publications, 1998, 1998 (Special) : 276-287. doi: 10.3934/proc.1998.1998.276
[5]

Tetsuya Ishiwata, Shigetoshi Yazaki. A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2069-2090. doi: 10.3934/dcds.2014.34.2069
[6]

Tetsuya Ishiwata, Takeshi Ohtsuka. Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5261-5295. doi: 10.3934/dcdsb.2019058
[7]

Francis Michael Russell, J. C. Eilbeck. Persistent mobile lattice excitations in a crystalline insulator. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1267-1285. doi: 10.3934/dcdss.2011.4.1267
[8]

Annalisa Malusa, Matteo Novaga. Crystalline evolutions in chessboard-like microstructures. Networks & Heterogeneous Media, 2018, 13 (3) : 493-513. doi: 10.3934/nhm.2018022
[9]

Ken Shirakawa. Stability analysis for phase field systems associated with crystalline-type energies. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 483-504. doi: 10.3934/dcdss.2011.4.483
[10]

Ke Xu, M. Gregory Forest, Xiaofeng Yang. Shearing the I-N phase transition of liquid crystalline polymers: Long-time memory of defect initial data. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 457-473. doi: 10.3934/dcdsb.2011.15.457
[11]

Ken Shirakawa. Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies. Conference Publications, 2009, 2009 (Special) : 697-707. doi: 10.3934/proc.2009.2009.697
[12]

Y. Goto, K. Ishii, T. Ogawa. Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature. Communications on Pure & Applied Analysis, 2005, 4 (2) : 311-339. doi: 10.3934/cpaa.2005.4.311
[13]

Oleksandr Misiats, Nung Kwan Yip. Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6379-6411. doi: 10.3934/dcds.2016076
[14]

Giovanni Bellettini, Matteo Novaga, Giandomenico Orlandi. Eventual regularity for the parabolic minimal surface equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5711-5723. doi: 10.3934/dcds.2015.35.5711
[15]

Kangsheng Liu, Xu Liu, Bopeng Rao. Eventual regularity of a wave equation with boundary dissipation. Mathematical Control & Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17
[16]

Benjamin Webb. Dynamics of functions with an eventual negative Schwarzian derivative. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1393-1408. doi: 10.3934/dcds.2009.24.1393
[17]

Alberto Ferrero, Filippo Gazzola, Hans-Christoph Grunau. Decay and local eventual positivity for biharmonic parabolic equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1129-1157. doi: 10.3934/dcds.2008.21.1129
[18]

Filippo Gazzola, Hans-Christoph Grunau. Eventual local positivity for a biharmonic heat equation in RN. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 83-87. doi: 10.3934/dcdss.2008.1.83
[19]

Chi Hin Chan, Magdalena Czubak, Luis Silvestre. Eventual regularization of the slightly supercritical fractional Burgers equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 847-861. doi: 10.3934/dcds.2010.27.847
[20]

Jinlong Bai, Xuewei Ju, Desheng Li, Xiulian Wang. On the eventual stability of asymptotically autonomous systems with constraints. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4457-4473. doi: 10.3934/dcdsb.2019127

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences