Existence and uniqueness of the motion by curvature of regular networks

Abstract

We prove existence and uniqueness of the motion by curvature of networks with triple junctions in ${\mathbb{R}}^d$ when the initial datum is of class $W_p^{2-2/p}$ and the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover, we investigate the regularisation effect due to the parabolic nature of the system. An application of the well-posedness is a new proof and a generalisation of the long-time behaviour result derived by Mantegazza et al. in 2004. Our study is motivated by an open question proposed in the 2016 survey from Mantegazza et al.: does there exist a unique solution of the motion by curvature of networks with initial datum being a regular network of class $C^2$? We give a positive answer.