Existence and Uniqueness of the Motion by Curvature of regular 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.