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

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.
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. 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. 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. 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, 13 (2000), 64. 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. 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. doi: 10.1007/BF03167495. Google Scholar

[7]

M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane,, Oxford, (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. 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., 4 (2011), 865. 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. 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. 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, 52 (1991), 321. Google Scholar

[13]

S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion,, Hokkaido Math. J., 30 (2001), 327. 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. 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. 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. 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. 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. 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, 13 (2000), 64. 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. 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. doi: 10.1007/BF03167495. Google Scholar

[7]

M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane,, Oxford, (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. 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., 4 (2011), 865. 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. 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. 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, 52 (1991), 321. Google Scholar

[13]

S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion,, Hokkaido Math. J., 30 (2001), 327. 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. 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. doi: 10.1051/m2an:1999164. Google Scholar

[1]

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

[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, 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

[4]

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

[5]

Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect. Networks & Heterogeneous Media, 2011, 6 (1) : 77-88. doi: 10.3934/nhm.2011.6.77

[6]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148

[7]

Doan The Hieu, Tran Le Nam. The classification of constant weighted curvature curves in the plane with a log-linear density. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1641-1652. doi: 10.3934/cpaa.2014.13.1641

[8]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061

[9]

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

[10]

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

[11]

Yaiza Canzani, Dmitry Jakobson, Igor Wigman. Scalar curvature and $Q$-curvature of random metrics. Electronic Research Announcements, 2010, 17: 43-56. doi: 10.3934/era.2010.17.43

[12]

Fabio Nicola. Remarks on dispersive estimates and curvature. Communications on Pure & Applied Analysis, 2007, 6 (1) : 203-212. doi: 10.3934/cpaa.2007.6.203

[13]

Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911

[14]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119

[15]

Yves Coudène, Barbara Schapira. Counterexamples in non-positive curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1095-1106. doi: 10.3934/dcds.2011.30.1095

[16]

Stefanella Boatto. Curvature perturbations and stability of a ring of vortices. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 349-375. doi: 10.3934/dcdsb.2008.10.349

[17]

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

[18]

Koichi Osaki, Hirotoshi Satoh, Shigetoshi Yazaki. Towards modelling spiral motion of open plane curves. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1009-1022. doi: 10.3934/dcdss.2015.8.1009

[19]

Jiří Minarčík, Masato Kimura, Michal Beneš. Comparing motion of curves and hypersurfaces in $ \mathbb{R}^m $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4815-4826. doi: 10.3934/dcdsb.2019032

[20]

G. Kamberov. Prescribing mean curvature: existence and uniqueness problems. Electronic Research Announcements, 1998, 4: 4-11.

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]