Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations

TL;DR

This paper analyzes the well-posedness and long-term behavior of a coupled fluid-structure system involving viscous fluid dynamics and nonlinear elastic shell equations, demonstrating existence of a global attractor under certain damping conditions.

Contribution

It establishes the generation of a semiflow for the coupled system and proves the existence of a compact global attractor considering rotational inertia and damping effects.

Findings

The coupled system generates a semiflow in an appropriate phase space.

Viscous dissipation regularizes the system allowing for inertia considerations.

Existence of a global attractor is proven under damping and external force conditions.

Abstract

We study well-posedness and asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) full von Karman shallow shell equations that accounts for both transversal and lateral displacements on a flexible part of the boundary. We also take into account rotational inertia of filaments of the shell. Out main result shows that the problem generates a semiflow in an appropriate phase space. The regularity provided by viscous dissipation in the fluid allows us to consider simultaneously both cases of presence inertia in the lateral displacements and its absence. Our second result states the existence of a compact global attractor for this semiflow in the case of presence of (rotational) damping in the transversal component and a particular structure of external forces.