Elastic flow of networks: short-time existence result

Anna Dall’Acqua*, Chun Chi Lin, Paola Pozzi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

In this paper we study the L2-gradient flow of the penalized elastic energy on networks of q-curves in Rn for q≥ 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 Hölder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.

Original languageEnglish
Pages (from-to)1299-1344
Number of pages46
JournalJournal of Evolution Equations
Volume21
Issue number2
DOIs
Publication statusAccepted/In press - 2020

Keywords

  • Elastic networks
  • Geometric evolution
  • Junctions
  • Short-time existence

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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