Elastic flow of networks: short-time existence result

TL;DR

This paper proves short-time existence of smooth solutions for the elastic flow of networks of curves with fixed endpoints and a movable junction, using an analytical approach to handle reparametrization issues.

Contribution

It introduces a novel analytical framework for the elastic flow of networks, establishing short-time existence by linking a parabolic system to the geometric evolution.

Findings

Existence of smooth solutions for short time intervals.

Development of an analytical approach to handle reparametrization freedom.

Application of Solonnikov's theory and fixed point theorem to geometric flow.

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.