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On the curve diffusion flow of plane curves

by Wheeler, Glen.

Series: 2011-19, Preprints

35G25 Initial value problems for nonlinear higher-order equations

In this paper we consider the steepest descent H-1 gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. We prove that under the flow closed or complete periodic curves with initial data close to a multiply-covered circle or straight line respectively. In the first case we also provide an estimate for the time during which the curvature is not strictly positive. As an application of these results we obtain a rigidity statement for immersed planar curves.

curve diffusion, plane curves