## M. Gößwein - J. Menzel - A. Pluda

# 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.

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