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

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 equation 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. Our 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.

Dynamics of Quasi-Stable Dissipative Systems

“…The same idea was recently applied in CHUESHOV/KOLBASIN [49] (see also Chapter 5) for the analysis of long-time dynamics in a degenerate hyperbolic-type model. The quantum Zakharov system (see CHUESHOV [43] and the references therein) and several classes of fluid-structure interaction models (see, e.g., CHUESHOV [45,46] and CHUESHOV/RYZHKOVA [68,69]) also demonstrate some applications of quasistability idea. Definition 3.4.1.…”

A linearized viscous, compressible flow-plate interaction with non-dissipative coupling

“…As with many fluid-structure interactions, the critical issue in [15] is the appearance of ill-defined interface traces. In the incompressible case, one can recover negative Sobolev trace regularity for the pressure at the interface via properties of the Stokes' operator [12,13,19,21]. However, in the viscous compressible case this is no longer true.…”

“…For incompressible flows, the analysis typically involves two unknowns: velocity and pressure (e.g., [12,13,19,21]). One must solve both conservation of mass and linear momentum equations, with the fluid density constant.…”

“…This problem is motivated by the main lines of the recent work of I. Chueshov et al [15,[19][20][21] on various types of 3-D/2-D fluid-structure interactions (as considered here). In these papers, the authors address various combinations of viscous compressible and incompressible fluid dynamics in 3-D domains (sometimes unbounded but tubular [20]) linearized about the steady flow state U ≡ 0, and coupled to different types of elastic dynamics at the interface (2-D in-plane elasticity [22], von Karman [21], or even full von Karman [19]). The Galerkin approach is utilized in constructing solutions, and dynamical systems techniques are used to obtain long-time behavior results for these systems.…”

Rational rates of uniform decay for strong solutions to a fluid-structure PDE system

“…It is here important to emphasize that the FSI PDE models which are considered in the above-mentioned references are structurally quite different -in terms of their respective coupling mechanisms -than the present one under consideration. For the well-posedness and long-time behavior analysis of other closely related fluid-plate PDE models, we refer the reader to [18] and [15].…”