On equations of continuity and transport type on metric graphs and fractals

TL;DR

This paper establishes well-posedness for first order continuity and transport equations on metric graphs and fractals, introducing new analytical tools and boundary conditions to handle non-uniqueness issues.

Contribution

It provides the first well-posedness results for scalar first order equations on fractal spaces using boundary quadruples and new domain characterizations.

Findings

Proves well-posedness for divergence-free vector fields on fractals and graphs.

Introduces a new integration by parts formula considering vector fields and loops.

Analyzes duality and metric graph approximations with periodic boundary conditions.

Abstract

We study first order equations of continuity and transport type on metric spaces of martingale dimension one, including finite metric graphs, p.c.f. self-similar sets and classical Sierpi\'nski carpets. On such spaces solutions of the continuity equation in the weak sense are generally non-unique. We use semigroup theory to prove a well-posedness result for divergence free vector fields and under suitable loop and boundary conditions. It is the first well-posedness result for first order equations with scalar valued solutions on fractal spaces. A key tool is the concept of boundary quadruples recently introduced by Arendt, Chalendar and Eymard. To exploit it, we prove a new domain characterization for the relevant first order operator and a novel integration by parts formula, which takes into account the given vector field and the loop structure of the space. We provide additional results on duality and on metric graph approximations in the case of periodic boundary conditions.