Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gromov-Hausdorff stability of global attractors of damped wave equations under perturbations of the domain

Ngoctu Bui, Jihoon Lee

TL;DR

This work addresses how global attractors for a damped wave equation respond to perturbations of the domain, employing the Gromov-Hausdorff framework to formalize stability. By parameterizing domain perturbations with $d_{C^2}(h,1_\\Omega)=\delta$ and transporting solutions via coordinate maps, the authors establish residual continuity of the attractor map $\\mathcal{G}(h)=\\mathcal{A}_{\\delta}$ and residual Gromov-Hausdorff stability for the induced dynamical systems on the attractors. The core contributions include key lemmas showing convergence of perturbed evolutions under diffeomorphisms and a two-step analysis proving equicontinuity and small-time stability, which together yield robust attractor stability under geometric perturbations. The results advance understanding of long-time behavior for hyperbolic PDEs on varying domains and have implications for the reliability of attractor-based descriptions in geometrically perturbed settings.

Abstract

In this paper, we will make use of the Gromov-Hausdorff distance between compact metric spaces to establish the continuous dependence and the Gromov-Hausdorff stability of global attractors for damped wave equations under perturbations of the domain.

Gromov-Hausdorff stability of global attractors of damped wave equations under perturbations of the domain

TL;DR

This work addresses how global attractors for a damped wave equation respond to perturbations of the domain, employing the Gromov-Hausdorff framework to formalize stability. By parameterizing domain perturbations with and transporting solutions via coordinate maps, the authors establish residual continuity of the attractor map and residual Gromov-Hausdorff stability for the induced dynamical systems on the attractors. The core contributions include key lemmas showing convergence of perturbed evolutions under diffeomorphisms and a two-step analysis proving equicontinuity and small-time stability, which together yield robust attractor stability under geometric perturbations. The results advance understanding of long-time behavior for hyperbolic PDEs on varying domains and have implications for the reliability of attractor-based descriptions in geometrically perturbed settings.

Abstract

In this paper, we will make use of the Gromov-Hausdorff distance between compact metric spaces to establish the continuous dependence and the Gromov-Hausdorff stability of global attractors for damped wave equations under perturbations of the domain.
Paper Structure (3 sections, 7 theorems, 67 equations)

This paper contains 3 sections, 7 theorems, 67 equations.

Table of Contents

  1. INTRODUCTION
  2. PROOF OF THEOREM \ref{['thm1']}
  3. PROOF OF THEOREM \ref{['thm2']}

Key Result

Theorem 1.1

The map $\mathcal{G}: \mathcal{P}(\Omega) \rightarrow\mathcal{M}$ given by $\mathcal{G}(h) = \mathcal{A}_{\delta}$ is residually continuous, where $d_{C^2}(h, 1_\Omega)=\delta$ and $\mathcal{A}_{\delta}$ is the global attractor of perturbed.

Theorems & Definitions (13)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof : Proof of Theorem $\ref{['thm1']}$
  • ...and 3 more