Elastic flow of networks: short-time existence result

Anna Dall'Acqua; Chun-Chi Lin; Paola Pozzi

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arXiv · 1912.09626

Elastic flow of networks: short-time existence result

Anna Dall'Acqua·Chun-Chi Lin·Paola Pozzi

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Abstract

In this paper we study the $L^2$-gradient flow of the penalized elastic energy on networks of $q$-curves in $\R^{n}$ for $q \geq 3$. Each curve is fixed at one end-point and at the other is joint to the other curves at a movable $q$-junction. For this geometric evolution problem with natural boundary condition we show the existence of smooth solutions for a (possibly) short interval of time. Since the geometric problem is not well-posed, due to the freedom in reparametrization of curves, we consider a fourth-order non-degenerate parabolic quasilinear system, called the analytic problem, and show first a short-time existence result for this parabolic system. The proof relies on applying Solonnikov's theory on linear parabolic systems and Banach fixed point theorem in proper H\"{o}lder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.

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Anna Dall'Acqua, Chun-Chi Lin, Paola Pozzi. 2019-12-20. Elastic flow of networks: short-time existence result. https://arxiv.org/abs/1912.09626

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