and [@MRW]). To apply this method we need the property ${{\gamma}}>0$, i.e., assume a presence of rotational damping in the transversal component of displacement. The question whether the system under consideration demonstrates compact long-time behavior without mechanical damping in the shell component is still open. In contrast we note that the existence of global attractors in the models considered in [@Chu_2010; @ChuRyz2011] [*does not require*]{} any mechanical damping and compact asymptotic dynamics of the corresponding system is guaranteed by viscous dissipation of fluid. One of the reasons for this is that the models in [@Chu_2010; @ChuRyz2011] do not include higher order (rotational type) inertial terms and thus the finiteness of the [*full*]{} dissipation integral for the displacement follows from the viscosity of the fluid via the compatibility condition in .

The paper organized as follows. In Section \[sec:pre\] we provide some preliminary material related to Sobolev spaces and the Stokes problem. In Section \[sec:WP\] we state and prove our main well-posedness result. Section \[sec:asymp\] deals with long-time dynamics.

Preliminaries {#sec:pre}
=============

In this section we introduce Sobolev type spaces we need and provide with some results concerning the Stokes problem.

Spaces and notations
--------------------

To introduce Sobolev spaces we follow 