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Motion of spiral-shaped polygonal curves by nonlinear crystalline motion with a rotating tip motion. (English) Zbl 1349.34040

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.

MSC:

34A34 Nonlinear ordinary differential equations and systems
53A04 Curves in Euclidean and related spaces
82D25 Statistical mechanics of crystals