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

Alexander Dobrick, Florian G. Martin

TL;DR

Addresses well-posedness and equilibrium behavior of buffered transport on infinite networks by casting the problem as a $C_0$-semigroup evolution on an AL-space. The authors develop a Desch–Schappacher-type perturbation theory for non-analytic semigroups and apply it to obtain generation by the perturbed boundary operator. They prove irreducibility and link long-time behavior to the spectrum of the transposed adjacency matrix $\\mathbb{B}$, showing convergence to equilibrium under the spectral condition $1 \in \sigma_p(\\mathbb{B})$, with strong results for finite graphs yielding convergence in norm. The work provides a rigorous framework connecting network topology, buffer dynamics, and semigroup methods to guarantee convergence to steady states in buffered infinite networks.

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.

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

TL;DR

Addresses well-posedness and equilibrium behavior of buffered transport on infinite networks by casting the problem as a -semigroup evolution on an AL-space. The authors develop a Desch–Schappacher-type perturbation theory for non-analytic semigroups and apply it to obtain generation by the perturbed boundary operator. They prove irreducibility and link long-time behavior to the spectrum of the transposed adjacency matrix , showing convergence to equilibrium under the spectral condition , with strong results for finite graphs yielding convergence in norm. The work provides a rigorous framework connecting network topology, buffer dynamics, and semigroup methods to guarantee convergence to steady states in buffered infinite networks.

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 -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.
Paper Structure (5 sections, 23 theorems, 69 equations)

This paper contains 5 sections, 23 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Buffered transport problem on a graph
  3. Relatively compact perturbations of semigroup generators
  4. Well-posedness of the transport problem
  5. Long-term behaviour of the flow

Key Result

Proposition 3.1

Let $A$ be a densely defined linear operator on a Banach space $X$ and let $B \colon D(A) \to X$ be a compact operator. Suppose that there exists $\lambda_0 > 0$ such that $[\lambda_0,\infty) \subseteq \rho(A)$ and If $\lambda > \lambda_0$ is sufficiently large, then $\lambda \in \rho(A + B)$ and where the series is convergent in $\mathcal{L}(D(A))$.

Theorems & Definitions (50)

  • Proposition 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Proposition \ref{['prop:resolvent-set-perturbation']}
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • ...and 40 more