Singular limit and long-time dynamics of Bresse systems

TL;DR

This paper studies the Bresse system modeling arched beams, showing it converges to the Timoshenko system as curvature approaches zero, and analyzes its long-term behavior including attractors with nonlinear damping.

Contribution

It establishes the singular limit of the Bresse system to the Timoshenko system and investigates the existence and properties of global and exponential attractors under nonlinear damping.

Findings

Bresse system converges to Timoshenko system as curvature tends to zero.

Existence of smooth global attractors with finite fractal dimension.

Comparison of attractors shows upper semicontinuity as curvature approaches zero.

Abstract

The Bresse system is a valid model for arched beams which reduces to the classical Timoshenko system when the arch curvature $\ell=0$. Our first result shows the Timoshenko system as a singular limit of the Bresse system as $\ell \to 0$. The remaining results are concerned with the long-time dynamics of Bresse systems. In a general framework, allowing nonlinear damping and forcing terms, we prove the existence of a smooth global attractor with finite fractal dimension and exponential attractors as well. We also compare the Bresse system with the Timoshenko system, in the sense of upper semicontinuity of their attractors as $\ell \to 0$.