Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regularity and stability of two coupled Euler-Bernoulli equations with a localized singular structural damping

K. Ammari, F. Hassine, L. Tebou

Abstract

This paper is concerned with the study of regularity and stability properties of two Euler-Bernoulli beam equations with localized singular damping. Under suitable regularity assumptions on the damping coefficient, we establish Gevrey regularity for the semigroup generated by the associated operator. Furthermore, for a broader class of damping mechanisms, including less regular damping, we derive uniform stability result. These findings provide a detailed description of the long-term behavior of the corresponding dynamical systems.

Regularity and stability of two coupled Euler-Bernoulli equations with a localized singular structural damping

Abstract

This paper is concerned with the study of regularity and stability properties of two Euler-Bernoulli beam equations with localized singular damping. Under suitable regularity assumptions on the damping coefficient, we establish Gevrey regularity for the semigroup generated by the associated operator. Furthermore, for a broader class of damping mechanisms, including less regular damping, we derive uniform stability result. These findings provide a detailed description of the long-term behavior of the corresponding dynamical systems.
Paper Structure (4 sections, 5 theorems, 147 equations)

This paper contains 4 sections, 5 theorems, 147 equations.

Table of Contents

  1. Introduction and statements of main results
  2. Well-posedness and strong stability
  3. Proof of Theorem \ref{['reg']}
  4. Proof of Theorem \ref{['expstab']}

Key Result

Theorem 1.1

Suppose now that $\omega$ satisfies the geometric constraint (GC). Further, assume that the function $a$ lies in $C^2(\bar{\Omega})$, vanishes in $\Omega\setminus\omega$, and satisfies: where $D^2a(x)$ denotes the Hessian matrix of $a$ at $x$. Then, the semigroup $(S(t))_{t\geq0}$ is of Gevrey class $s$ for every $s>5/2$, as its resolvent satisfies the following estimate Therefore, the semigrou

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 4.1