# Existence and Uniqueness of the Motion by Curvature of regular networks

created by pluda on 10 Mar 2020
modified on 22 Mar 2020

[BibTeX]

Preprint

Inserted: 10 mar 2020
Last Updated: 22 mar 2020

Year: 2020

Abstract:

We prove existence and uniqueness of the motion by curvature of networks in $\mathbb{R}^n$ when the initial datum is of class $W^{2-\frac{2}{p}}_p$ with $p\in(3,\infty)$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in "Motion by Curvature of Planar Networks" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our study is motivated by an open question proposed in "Evolution of Networks with Multiple Junctions " by Mantegazza-Novaga-Pluda-Schulze: does there exist a unique solution of the motion by curvature of networks with initial datum a regular network of class $C^2$? We give a positive answer.