Flows on Metric Graphs with General Boundary Conditions

TL;DR

This paper investigates the generation of $C_0$-semigroups by first order differential operators on metric graphs with general boundary conditions, providing characterizations and applications to transport equations.

Contribution

It offers a new characterization of semigroup generation via boundary condition matrices and applies these results to transport equations on non-compact metric graphs.

Findings

Characterization of semigroup generation through boundary matrices

Conditions for well-posedness of transport equations on metric graphs

Application to non-compact graph boundary value problems

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.