Motion of spirals by crystalline curvature

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1. [1] R. Almgren, Crystalline Saffman-Taylor fingers. SIAM J. Appl. Math. 55 (1995) 1511-1535. Zbl0838.76094MR1358787
2. [2] S. Angenent and M.E. Gurtin, Multiphase thermodynamics with interfacial structure 2. Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108 (1989) 323-391. Zbl0723.73017MR1013461
3. [3] C.M. Elliot, A.R. Gardiner and R. Schätzle, Crystalline curvature flow of a graph in a variational setting. Adv. Math. Sci. Appl. 8 (1998) 425-460. Zbl0959.35168MR1623315
4. [4] T. Fukui and Y. Giga, Motion of a graph by nonsmooth weighted curvature, in World Congress of Nonlinear Analysis '92, Vol. I, V. Lakshmikantham Ed., Walter de Gruyter, Berlin (1996) 47-56. Zbl0860.35061MR1389060
5. [5] M.E. Gage, On the size of the blow-up set for a quasilinear parabolic equation. Contemp. Math. 127 (1992) 41-58. Zbl0770.35029MR1155408
6. [6] M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal. 141 (1998) 117-198. Zbl0896.35069MR1615520
7. [7] Y. Giga and M.E. Gurtin, A comparison theorem for crystalline evolution in the plane. Quart. Appl Math. 54 (1996) 727-737. Zbl0862.35047MR1417236
8. [8] Y. Giga, M.E. Gurtin and J. Matias, On the dynamics of crystalline motion. Japan J. Indust, Appl. Math. 15 (1998) 7-50. Zbl1306.74038MR1610305
9. [9] P.M. Girão, Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature. SIAM J. Numer. Anal. 32 (1995) 886-899. Zbl0830.65150MR1335660
10. [10] P.M. Girão and R.V. Kohn, Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature. Numer. Math. 67 (1994) 41-70. Zbl0791.65063MR1258974
11. [11] M.E. Gurtin, Thermomechanics of evolving phase boundaries in the plane. Oxford, Clarendon Press (1993). Zbl0787.73004MR1402243
12. [12] R. Ikota, N. Ishimura and T. Yamaguchi, On the structure of steady solutions for the kinematic model of spiral waves in excitable media. Japan J. Indust. Appl. Math. 15 (1998) 317-330. Zbl0904.92002MR1629103
13. [13] K. Ishii and H.M. Soner, Regularity and convergence of crystalline motion. Preprint (1996). Zbl0963.35082MR1646732
14. [14] N. Ishimura, Shape of spiral. Tôhoku Math. J. 50 (1998) 197-202. Zbl0915.35048MR1622050
15. [15] W. Jahnke and A.T. Winfree, A survey of spiral-wave behaviors in the oregonator model. Internat J, Bifur. Chaos 1 (1991). 445-466. Zbl0900.92066MR1120210
16. [16] T. Kuroda, Kessyou ha ikiteiru (Crystal is alive). Science Sya, Tokyo (1984). 
17. [17] A.S. Mikhailov, V.A. Davydov and A.S. Zykov, Complex dynamics of spiral waves and motion of curves. Pkysica D 70 (1994) 1-39. Zbl0807.58043MR1257848
18. [18] A. Ookawa, Kessyou seicyo (Crystal Growth). Syokabo, Tokyo (1977). 
19. [19] A.R. Roosen and J.E. Taylor, Modeling crystal growth in a diffusion field using fully faceted interfaces. J. Comput. Phys. 114 (1994) 113-128. Zbl0805.65128MR1286190
20. [20] P. Rybka, A quasi-steady approximation to an integro-differential model of interface motion. Appl Anal. 56 (1995) 19-34. Zbl0832.35155MR1378009
21. [21] P. Rybka, A crystalline motion: uniqueness and geometric properties. SIAM J. Appl. Math. 57 (1997) 53-72. Zbl0870.35129MR1429377
22. [22] I. Sunagawa, K. Narita, P. Bennema and B. van der Hoek, Observation and interpretation of eccentric growth spirals. J. Crystal Growth 42 (1977) 121-126. 
23. [23] J.E. Taylor, Crystalline variational problem. Bull. Amer. Math. Soc. 84 (1978) 568-588. Zbl0392.49022MR493671
24. [24] J.E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, in Differential Geometry (A symposium in honour of Manfredo do Carmo) B, Lawson and K, Tenenblat Eds., Longman, Essex (1991) 321-336. Zbl0725.53011MR1173051
25. [25] J.E. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points. Proc. Sympos. Pure. Math. 54 (1993) 417-438. Zbl0823.49028MR1216599
26. [26] J.E. Taylor, Surface motion due to crystalline surface energy gradient flows, in Elliptic and Parabolic Methods in Geometry, B. Chow, R. Gulliver, S. Levy, J. Sulliva and A.K. Peters Eds., Massachusetts (1996) 145-162. Zbl0915.49024MR1417953
27. [27] J.J. Tyson and J.P. Keener, Singular perturbation theory of traveling waves in excitable media. Physica D 32 (1988) 327-361. Zbl0656.76018MR980194
28. [28] T. Ushijima and S. Yazaki, Convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature v = kα. Preprint (1997). Zbl0946.65071MR1740769