Submitted Paper
Inserted: 28 sep 2023
Last Updated: 11 apr 2024
Year: 2023
Abstract:
We consider a curve with boundary points free to move on a line in $\mathbb R^2$, which evolves by the $L^2$--gradient flow of the elastic energy, that is a linear combination of the Willmore and the length functional. For such planar evolution problem we study the short and long--time existence. Once we establish under which boundary conditions the PDE's system is well--posed (in our case the Navier boundary conditions), employing the Solonnikov theory for linear parabolic systems in H\"older space, we show that there exists a unique flow in a maximal time interval $[0,T)$. Then, using energy methods we prove that the maximal time is actually $T= + \infty$.
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