Well-posedness and long-term behaviour of buffered flows in infinite networks

TL;DR

This paper analyzes the well-posedness and long-term dynamics of buffered flow transport on infinite networks, employing semigroup theory and spectral analysis to establish convergence to equilibrium.

Contribution

It introduces a novel application of $C_0$-semigroup perturbation theory to infinite network flows with buffering, and characterizes their asymptotic behaviour.

Findings

Proves well-posedness using dispersive semigroup theory.

Establishes conditions for strong and uniform convergence to equilibrium.

Shows irreducibility of the semigroup under strong connectivity and spectral conditions.

Abstract

We consider a transport problem on an infinite metric graph and discuss its well-posedness and long-term behaviour under the condition that the mass flow is buffered in at least one of the vertices. In order to show the well-posedness of the problem, we employ the theory of $C_0$-semigroups and prove a Desch--Schappacher type perturbation theorem for dispersive semigroups. Investigating the long-term behaviour of the system, we prove irreducibility of the semigroup under the assumption that the underlying graph is strongly connected and an additional spectral condition on its adjacency matrix. Moreover, we employ recent results about the convergence of stochastic semigroups that dominate a kernel operator to prove that the solutions converge strongly to equilibrium. Finally, we prove that the solutions converge uniformly under more restrictive assumptions.