Existence and Uniqueness of the Motion by Curvature of regular networks

TL;DR

This paper establishes the existence and uniqueness of curvature-driven motion for regular networks with triple junctions in Euclidean space, providing new insights into the regularization effects and answering an open question about solutions with smooth initial data.

Contribution

It proves existence and uniqueness of network motion by curvature for initial data in a specific Sobolev space, and confirms solutions exist for smooth initial networks, addressing an open problem.

Findings

Existence and uniqueness of solutions for initial data in W^{2-2/p}_p.

Regularization effects due to the parabolic nature of the system.

Positive answer to the open question for C^2 initial networks.

Abstract

We prove existence and uniqueness of the motion by curvatureof networks in $\mathbb{R}^n$ when the initial datum is of class $W^{2-\frac{2}{p}}_p$, 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.