Quantum graphs with mixed dynamics: the transport/diffusion case

TL;DR

This paper introduces a new class of PDEs on metric graphs combining diffusion and transport processes, providing conditions for well-posedness and applicability to systems with boundary delays.

Contribution

It develops a framework for PDEs on metric graphs with mixed dynamics and nonlocal boundary couplings, expanding the analysis of such systems.

Findings

Established conditions for contractive semigroup generation.

Applicable to systems with boundary delays.

Unified treatment of diffusion and transport on graphs.

Abstract

We introduce a class of partial differential equations on metric graphs associated with mixed evolution: on some edges we consider diffusion processes, on other ones transport phenomena. This yields a system of equations with possibly nonlocal couplings at the boundary. We provide sufficient conditions for these to be governed by a contractive semigroup on a Hilbert space naturally associated with the system. We show that our setting is also adequate to discuss specific systems of diffusion equations with boundary delays.