Flows on Metric Graphs with General Boundary Conditions

Abstract

In this note we study the generation of $C_0$-semigroups by first order differential operators on ${\mathrm{L}}^p({\mathbb{R}}_{+},{\mathbb{C}}^{\ell })\times {\mathrm{L}}^p([0,1],{\mathbb{C}}^m)$ with general boundary conditions. In many cases we are able to characterize the generation property in terms of the invertibility of a matrix associated to the boundary conditions. The abstract results are used to study well-posedness of transport equations on non-compact metric graphs.