Regularity of boundary traces for a fluid-solid interaction model
Francesca Bucci Irena Lasiecka
Discrete & Continuous Dynamical Systems - S 2011, 4(3): 505-521 doi: 10.3934/dcdss.2011.4.505
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.
keywords: trace regularity optimal control problems. Fluid-solid interactions
Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations
Francesca Bucci Igor Chueshov
Discrete & Continuous Dynamical Systems - A 2008, 22(3): 557-586 doi: 10.3934/dcds.2008.22.557
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.
keywords: nonlinear damping Coupled PDE system global attractor finite fractal dimension critical exponent.
Global attractor for a composite system of nonlinear wave and plate equations
Francesca Bucci Igor Chueshov Irena Lasiecka
Communications on Pure & Applied Analysis 2007, 6(1): 113-140 doi: 10.3934/cpaa.2007.6.113
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.
keywords: wave-plate model critical exponents. nonlinear damping finite fractal dimension Global attractor

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