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Theory And Applications Of One-Sided Coupled Operator Matrices

Marjeta Kramar, Delio Mugnolo, Rainer Nagel

TL;DR

The paper surveys the theory of one-sided coupled operator matrices (osc) as a unified framework for abstract initial-value problems with unbounded boundary feedbacks.It develops an osc framework, establishes a spectral transfer principle, and provides generator and positivity criteria for the resulting block-operator systems.It then applies the theory to initial-boundary value problems, including diffusion–transport systems with dynamical boundary conditions and a wave equation, proving well-posedness and, in many cases, exponential stability.The results yield practical tools for analyzing complex PDEs with boundary feedback by reducing them to tractable operator-matrix conditions.

Abstract

The theory of one-sided coupled operator matrices, recently introduced by K.-J. Engel, is an abstract framework for concrete initial value problems and allows complete information on well-posedness, and stability of solutions. These notes are meant as a survey on this rich theory, with a particular stress on applications to initial-boundary value problems with unbounded boundary feedbacks. A diffusion-transport system with dynamical boundary conditions is discussed, and its well-posedness and various other properties are investigated. As a by-product, the well-posedness of a wave equation with dynamical boundary condition is also obtained.

Theory And Applications Of One-Sided Coupled Operator Matrices

TL;DR

The paper surveys the theory of one-sided coupled operator matrices (osc) as a unified framework for abstract initial-value problems with unbounded boundary feedbacks.It develops an osc framework, establishes a spectral transfer principle, and provides generator and positivity criteria for the resulting block-operator systems.It then applies the theory to initial-boundary value problems, including diffusion–transport systems with dynamical boundary conditions and a wave equation, proving well-posedness and, in many cases, exponential stability.The results yield practical tools for analyzing complex PDEs with boundary feedback by reducing them to tractable operator-matrix conditions.

Abstract

The theory of one-sided coupled operator matrices, recently introduced by K.-J. Engel, is an abstract framework for concrete initial value problems and allows complete information on well-posedness, and stability of solutions. These notes are meant as a survey on this rich theory, with a particular stress on applications to initial-boundary value problems with unbounded boundary feedbacks. A diffusion-transport system with dynamical boundary conditions is discussed, and its well-posedness and various other properties are investigated. As a by-product, the well-posedness of a wave equation with dynamical boundary condition is also obtained.

Paper Structure

This paper contains 9 sections, 25 theorems, 112 equations.

Table of Contents

  1. INTRODUCTION
  2. VOLTERRA AND DELAY DIFFERENTIAL EQUATIONS
  3. GENERAL FRAMEWORK
  4. CHARACTERIZATION OF ONE-SIDED COUPLED OPERATOR MATRICES
  5. SPECTRAL THEORY FOR ONE-SIDED COUPLED OPERATOR MATRICES
  6. ONE-SIDED COUPLED GENERATORS OF STRONGLY CONTINUOUS SEMIGROUPS
  7. ONE-SIDED COUPLED OPERATOR MATRICES AND POSITIVITY
  8. APPLICATION TO INITIAL--BOUNDARY VALUE PROBLEMS
  9. A DIFFUSION–TRANSPORT SYSTEM WITH DYNAMICAL BOUNDARY CONDITIONS

Key Result

Proposition 4.1

Let $X$, $Y$ be Banach spaces and $(\mathcal{A},D(\mathcal{A}))$ be a linear operator on $\mathcal{X}: =X\times Y$. Define the operators $B$ and $D$ by If $\mathcal{A}$ is invertible and $\mathcal{A}^{-1}=$, then the following properties are equivalent.

Theorems & Definitions (62)

  • Definition 3.2
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 5.1
  • proof
  • Remark 5.2
  • Theorem 6.1
  • Remark 6.2
  • ...and 52 more