A global attractor for a fluid--plate interaction model accounting only for longitudinal deformations of the plate

TL;DR

This paper proves the existence of a finite-dimensional global attractor for a coupled fluid-plate system modeling blood flow in arteries, showing fluid viscosity alone stabilizes the system without mechanical damping.

Contribution

It establishes the existence of a finite-dimensional global attractor for a fluid-plate interaction model with only longitudinal deformations, without requiring damping in the plate.

Findings

Existence of a compact global attractor of finite dimension.

The linearized system generates an exponentially stable semigroup.

Fluid viscosity alone ensures system stabilization.

Abstract

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. The main peculiarity of the model is the assumption that the transversal displacements of the plate are negligible relative to in-plane displacements. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. We also show that the corresponding linearized system generates exponentially stable $C_0$-semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system.