The gradient flow of the potential energy on the space of arcs

@article{Shi2019TheGF,
  title={The gradient flow of the potential energy on the space of arcs},
  author={Wenhui Shi and Dmitry Vorotnikov},
  journal={Calculus of Variations and Partial Differential Equations},
  year={2019},
  volume={58},
  pages={1-27}
}
  • Wenhui Shi, D. Vorotnikov
  • Published 2 February 2017
  • Mathematics
  • Calculus of Variations and Partial Differential Equations
We study the gradient flow of the potential energy on the infinite-dimensional Riemannian manifold of spatial curves parametrized by the arc length, which models overdamped motion of a falling inextensible string. We prove existence of generalized solutions to the corresponding nonlinear evolutionary PDE and their exponential decay to the equilibrium. We also observe that the system admits solutions backwards in time, which leads to non-uniqueness of trajectories. 
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Correction to: Uniformly Compressing Mean Curvature Flow
A Correction to this paper has been published: 10.1007/s12220-018-00104-z

References

SHOWING 1-10 OF 29 REFERENCES
THE MOTION OF ELASTIC PLANAR CLOSED CURVES UNDER THE AREA-PRESERVING CONDITION
We consider the motion of an elastic closed curve with constant enclosed area. This motion is governed by a system involving fourth order parabolic equations. We shall prove that this system has a
Uniformly Compressing Mean Curvature Flow
Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints
ON THE MOTION OF A CURVE TOWARDS ELASTICA
We consider a non-linear 4-th order parabolic equation derived from bending energy of wires in the 3-dimensional Euclidean space. We show that a solution exists for all time, and converges to an
On the curve straightening flow of inextensible, open, planar curves
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
On the curve straightening flow of inextensible, open, planar curves
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
Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence
Convergence of the penalty method applied to a constrained curve straightening flow
We apply the penalty method to the curve straightening flow of inextensible planar open curves generated by the Kirchhoff bending energy. Thus we consider the curve straightening flow of extensible
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