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DCDS-S

We consider a mathematical model for the interactions of an elastic body fully immersed
in a viscous, incompressible fluid.
The corresponding composite PDE system comprises a linearized Navier-Stokes system
and a dynamic system of elasticity; the coupling takes place on the interface between the two regions occupied by the fluid and the solid, respectively.
We specifically study the regularity of boundary traces (on the interface)
for the fluid velocity field.
The obtained trace regularity theory for the fluid component of the system-of
interest in its own right-establishes, in addition, solvability of the associated optimal (quadratic) control problems on a finite time interval, along with well-posedness of the corresponding operator Differential Riccati equations.
These results complement the recent advances in the PDE analysis and control of the
Stokes-Lamé system.

DCDS

We prove the existence of a compact, finite dimensional, global attractor
for a coupled PDE system comprising a nonlinearly damped semilinear wave
equation and a nonlinear system of thermoelastic plate equations,
without any mechanical (viscous or structural) dissipation in the
plate component.
The plate dynamics is modelled following Berger's approach; we investigate
both cases when rotational inertia is included into the model and when it
is not.
A major part in the proof is played by an estimate--known as
stabilizability estimate--which shows that the difference of any
two trajectories can be exponentially stabilized to zero, modulo a
compact perturbation.
In particular, this inequality yields bounds for the attractor's fractal
dimension which are independent of two key parameters, namely $\gamma$
and $\kappa$, the former related to the presence of rotational inertia
in the plate model and the latter to the coupling terms.
Finally, we show the upper semi-continuity of the attractor with
respect to these parameters.

CPAA

We prove the existence of a compact, finite dimensional, global attractor
for a system of strongly coupled wave and plate equations
with nonlinear dissipation and forces.
This kind of models describes fluid-structure interactions.
Though our main focus is on the composite system of two partial differential
equations, the result achieved yields as well a new contribution to
the asymptotic analysis of either (uncoupled) equation.

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