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On the curve straightening flow of inextensible, open, planar curves

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Abstract

We consider the curve straightening flow of inextensible, open, planar curves generated by the Kirchhoff bending energy. It can be considered as a model for the motion of elastic, inextensible rods in a high friction regime. We derive governing equations, namely a semilinear fourth order parabolic equation for the indicatrix and a second order elliptic equation for the Lagrange multiplier. We prove existence and regularity of solutions, compute the energy dissipation, prove its coercivity and conclude convergence to equilibrium, namely to a straight curve, at an exponential rate.

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Authors and Affiliations

  1. Faculty of Mathematics, Vienna University, Austria

    D. B. Öelz

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  1. D. B. Öelz
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Correspondence to D. B. Öelz.

Additional information

This work has been started during a visit of the author at the Département de mathématiques et applications (DMA) of the ENS rue d’Ulm Paris within the DBASE project (Marie Curie Early Stage Training multi Site (mEST) of the EU, MEST-CT-2005-021122). Furthermore it has been supported by the Vienna Science and Technology Fund (WWTF) through the project ”How do cells move? Mathematical modelling of cytoskeletal dynamics and cell migration” of C. Schmeiser and V. Small.

The author would like to thank B. Perthame for suggesting the problem and for fruitful discussions and C. Schmeiser for helpful suggestions.

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Öelz, D.B. On the curve straightening flow of inextensible, open, planar curves. SeMA 54, 5–24 (2011). https://doi.org/10.1007/BF03322585

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