Existence and convergence of the length-preserving elastic flow of clamped curves

Abstract

We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative $L^2$-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic smoothing of the solution. Applying previous results on long time existence and proving a constrained Lojasiewicz-Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.

arXiv comment32 pages, final version. To appear in J. Evol. Equ

References in corpus (8)

• The Łojasiewicz-Simon gradient inequality for open elastic curves
• Convergence of elastic flows of curves into manifolds
• Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces
• The Lojasiewicz-Simon inequality for the elastic flow
• On the Lojasiewicz-Simon gradient inequality on submanifolds
• The volume-preserving Willmore flow
• The Willmore flow with prescribed isoperimetric ratio
• Asymptotic convergence of evolving hypersurfaces

Cited by in corpus (2)

• Li-Yau type inequality for curves in any codimension
• Dynamics of Elastic Wires: Preserving Area without Nonlocality