Geodesic
Motion of spiral-shaped polygonal curves by nonlinear crystalline motion with a rotating tip motion
Mathematica Bohemica, Tome 140 (2015) no. 2, pp. 111-119

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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.
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.
DOI : 10.21136/MB.2015.144318
Classification : 34A26, 34A34, 39A12, 74N05, 82D25
Keywords: curvature driven motion; crystalline curvature; spiral growth 
Ishiwata, Tetsuya. Motion of spiral-shaped polygonal curves by nonlinear crystalline motion with a rotating tip motion. Mathematica Bohemica, Tome 140 (2015) no. 2, pp. 111-119. doi: 10.21136/MB.2015.144318
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[1] Gurtin, M. E.: Thermomechanics of Evolving Phase Boundaries in the Plane. Oxford Mathematical Monographs Clarendon Press, Oxford (1993). | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

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