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Keywords:
curvature driven motion; crystalline curvature; spiral growth
curvature driven motion; crystalline curvature; spiral growth
Summary:
We consider a motion of spiral-shaped piecewise linear curves governed by a crystalline curvature flow with a driving force and a tip motion which is a simple model of a step motion of a crystal surface. We extend our previous result on global existence of a spiral-shaped solution to a linear crystalline motion for a power type nonlinear crystalline motion with a given rotating tip motion. We show that self-intersection of the solution curves never occurs and also show that facet extinction never occurs. Finally, we show that spiral-shaped solutions exist globally in time.
References:
[1] Gurtin, M. E.: Thermomechanics of Evolving Phase Boundaries in the Plane. Oxford Mathematical Monographs Clarendon Press, Oxford (1993). MR 1402243 | Zbl 0787.73004
[2] Ishiwata, T.: Motion of non-convex polygons by crystalline curvature and almost convexity phenomena. Japan J. Ind. Appl. Math. 25 (2008), 233-253. DOI 10.1007/BF03167521 | MR 2431681 | Zbl 1155.53033
[3] Ishiwata, T.: 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 | MR 3082855 | Zbl 1273.82076
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