Geodesic
On the tangential velocity arising in a crystalline approximation of evolving plane curves
Kybernetika, Tome 43 (2007) no. 6, pp. 913-918 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In a crystalline algorithm, a tangential velocity is used implicitly. In this short note, it is specified for the case of evolving plane curves, and is characterized by using the intrinsic heat equation.
In a crystalline algorithm, a tangential velocity is used implicitly. In this short note, it is specified for the case of evolving plane curves, and is characterized by using the intrinsic heat equation.
Classification : 34A26, 34A34, 35K65, 53A04, 53C44, 53C80, 65L20, 65M12, 65N12
Keywords: tangential velocity; intrinsic heat equation; crystalline algorithm; admissible polygonal curve 
@article{KYB_2007_43_6_a14,
     author = {Yazaki, Shigetoshi},
     title = {On the tangential velocity arising in a crystalline approximation of evolving plane curves},
     journal = {Kybernetika},
     pages = {913--918},
     year = {2007},
     volume = {43},
     number = {6},
     mrnumber = {2388404},
     zbl = {1139.53033},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2007_43_6_a14/}
}
TY  - JOUR
AU  - Yazaki, Shigetoshi
TI  - On the tangential velocity arising in a crystalline approximation of evolving plane curves
JO  - Kybernetika
PY  - 2007
SP  - 913
EP  - 918
VL  - 43
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/KYB_2007_43_6_a14/
LA  - en
ID  - KYB_2007_43_6_a14
ER  - 
%0 Journal Article
%A Yazaki, Shigetoshi
%T On the tangential velocity arising in a crystalline approximation of evolving plane curves
%J Kybernetika
%D 2007
%P 913-918
%V 43
%N 6
%U http://geodesic.mathdoc.fr/item/KYB_2007_43_6_a14/
%G en
%F KYB_2007_43_6_a14
Yazaki, Shigetoshi. On the tangential velocity arising in a crystalline approximation of evolving plane curves. Kybernetika, Tome 43 (2007) no. 6, pp. 913-918. http://geodesic.mathdoc.fr/item/KYB_2007_43_6_a14/

[1] Angenent S., Gurtin M. E.: Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108 (1989), 323–391 | DOI | MR

[2] Deckelnick K.: Weak solutions of the curve shortening flow. Calc. Var. Partial Differential Equations 5 (1997), 489–510 | DOI | MR | Zbl

[3] .Dziuk, G: Convergence of a semi discrete scheme for the curve shortening flow. Math. Models Methods Appl. Sci. 4 (1994), 589–606 | DOI | MR | Zbl

[4] Giga Y.: Anisotropic curvature effects in interface dynamics. Sūgaku 52 (2000), 113–127; English transl., Sugaku Expositions 16 (2003), 135–152 | MR

[5] Giga M.-H., Giga Y.: Crystalline and level set flow – convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane. GAKUTO Internat. Ser. Math. Sci. Appl. 13 (2000), 64–79 | MR | Zbl

[6] Gurtin M. E.: Thermomechanics of Evolving Phase Boundaries in the Plane. Clarendon Press, Oxford 1993 | MR | Zbl

[7] Hirota C., Ishiwata, T., Yazaki S.: Numerical study and examples on singularities of solutions to anisotropic crystalline curvature flows of nonconvex polygonal curves. In: Advanced Studies in Pure Mathematics (ASPM); Proc. MSJ-IRI 2005 “Asymptotic Analysis and Singularity”, Sendai 2005 (to appear) | MR | Zbl

[8] Hontani H., Giga M.-H., Giga, Y., Deguchi K.: Expanding selfsimilar solutions of a crystalline flow with applications to contour figure analysis. Discrete Appl. Math. 147 (2005), 265–285 | DOI | MR | Zbl

[9] Ishiwata T., Ushijima T. K., Yagisita, H., Yazaki S.: Two examples of nonconvex self-similar solution curves for a crystalline curvature flow. Proc. Japan Academy, Ser. A 80 (2004), 8, 151–154 | MR | Zbl

[10] Kimura M.: Accurate numerical scheme for the flow by curvature. Appl. Math. Letters 7 (1994), 69–73 | DOI | MR | Zbl

[11] Mikula K., Ševčovič D.: Solution of nonlinearly curvature driven evolution of plane curves. Appl. Numer. Math. 31 (1999), 191–207 | DOI | MR | Zbl

[12] Mikula K., Ševčovič D.: Evolution of plane curves driven by a nonlinear function of curvature and anisotropy. SIAM J. Appl. Math. 61 (2001), 1473–1501 | DOI | MR | Zbl

[13] Taylor J. E.: Constructions and conjectures in crystalline nondifferential geometry. In: Proc. Conference on Differential Geometry, Rio de Janeiro, Pitman Monographs Surveys Pure Appl. Math. 52 (1991), pp. 321–336, Pitman, London 1991 | MR | Zbl

[14] Taylor J. E.: Motion of curves by crystalline curvature, including triple junctions and boundary points. Diff. Geom.: Partial Diff. Eqs. on Manifolds (Los Angeles 1990), Proc. Sympos. Pure Math., 54 (1993), Part I, pp, 417–438, AMS, Providence | MR

[15] Taylor J. E., Cahn, J., Handwerker C.: Geometric models of crystal growth. Acta Metall. 40 (1992), 1443–1474 | DOI

[16] Ushijima T. K., Yagisita H.: Approximation of the Gauss curvature flow by a three-dimensional crystalline motion. In: Proc. Czech–Japanese Seminar in Applied Mathematics 2005; Kuju Training Center, Oita, Japan, September 15–18, 2005, COE Lecture Note 3, Faculty of Mathematics, Kyushu University (M. Beneš, M. Kimura, and T. Nakaki, eds.), 2006, pp. 139–145 | MR | Zbl

[17] Ushijima T. K., Yagisita H.: Convergence of a three-dimensional crystalline motion to Gauss curvature flow. Japan J. Indust. Appl. Math. 22 (2005), 443–459 | DOI | MR | Zbl

[18] Ushijima T. K., Yazaki S.: Convergence of a crystalline approximation for an area-preserving motion. Journal of Comput. and Appl. Math. 166 (2004), 427–452 | DOI | MR | Zbl

[19] Yazaki S.: Motion of nonadmissible convex polygons by crystalline curvature. Publ. Res. Inst. Math. Sci. 43 (2007), 155–170 | DOI | MR | Zbl