Abstract
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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Acknowledgements
The idea of this paper originated from conversations of the second author with Yann Brenier during a stay at ESI in Vienna. He would like to thank Yann Brenier for the inspiring discussions, Ulisse Stefanelli for the invitation to the thematic program “Nonlinear Flows” at ESI, and ESI for hospitality. The research was supported by CMUC (UID/MAT/00324/2019), funded by the Portuguese Government through FCT and co-funded by the ERDF through PT2020, and by FCT through Projects TUBITAK/0005/2014 and PTDC/MAT-PUR/28686/2017.
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Shi, W., Vorotnikov, D. The gradient flow of the potential energy on the space of arcs. Calc. Var. 58, 59 (2019). https://doi.org/10.1007/s00526-019-1524-1
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DOI: https://doi.org/10.1007/s00526-019-1524-1