# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [5] Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., 54 (1996), 727-737. [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. [7] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford, Clarendon Press, 1993. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [5] Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math., 54 (1996), 727-737. [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. [7] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford, Clarendon Press, 1993. [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. [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. [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. [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. [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. [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. [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. [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.
 [1] 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 [2] Tetsuya Ishiwata. Crystalline motion of spiral-shaped polygonal curves with a tip motion. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 53-62. doi: 10.3934/dcdss.2014.7.53 [3] Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390 [4] Tetsuya Ishiwata, Takeshi Ohtsuka. Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5261-5295. doi: 10.3934/dcdsb.2019058 [5] 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 [6] Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect. Networks and Heterogeneous Media, 2011, 6 (1) : 77-88. doi: 10.3934/nhm.2011.6.77 [7] Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389 [8] Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148 [9] Doan The Hieu, Tran Le Nam. The classification of constant weighted curvature curves in the plane with a log-linear density. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1641-1652. doi: 10.3934/cpaa.2014.13.1641 [10] Eszter Fehér, Gábor Domokos, Bernd Krauskopf. Tracking the critical points of curves evolving under planar curvature flows. Journal of Computational Dynamics, 2021, 8 (4) : 447-494. doi: 10.3934/jcd.2021017 [11] Annalisa Cesaroni, Valerio Pagliari. Convergence of nonlocal geometric flows to anisotropic mean curvature motion. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4987-5008. doi: 10.3934/dcds.2021065 [12] Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061 [13] 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 and Applied Analysis, 2005, 4 (2) : 311-339. doi: 10.3934/cpaa.2005.4.311 [14] Oleksandr Misiats, Nung Kwan Yip. Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6379-6411. doi: 10.3934/dcds.2016076 [15] 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 [16] Fabio Nicola. Remarks on dispersive estimates and curvature. Communications on Pure and Applied Analysis, 2007, 6 (1) : 203-212. doi: 10.3934/cpaa.2007.6.203 [17] Vittorio Martino. On the characteristic curvature operator. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911 [18] Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $\mathbb{R}^2$ with prescribed nonpositive curvature. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125 [19] Weimin Sheng, Caihong Yi. A class of anisotropic expanding curvature flows. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2017-2035. doi: 10.3934/dcds.2020104 [20] Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119

2020 Impact Factor: 2.425