A global attractor for a fluid--plate interaction model
I. D. Chueshov Iryna Ryzhkova
Communications on Pure & Applied Analysis 2013, 12(4): 1635-1656 doi: 10.3934/cpaa.2013.12.1635
We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) elastic plate equation for transversal displacement on a flexible flat part of the boundary. We show that this problem generates a semiflow on appropriate phase space. Our main result states the existence of a compact finite-dimensional global attractor for this semiflow. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system. To achieve the result we first study the corresponding linearized model and show that this linear model generates strongly continuous exponentially stable semigroup.
keywords: linearized 3D Navier--Stokes equations nonlinear plate Fluid--structure interaction finite-dimensional attractor.
Interaction of an elastic plate with a linearized inviscid incompressible fluid
I. D. Chueshov
Communications on Pure & Applied Analysis 2014, 13(5): 1759-1778 doi: 10.3934/cpaa.2014.13.1759
We prove well-posedness of energy type solutions to an interacting system consisting of the 3D linearized Euler equations and a (possibly nonlinear) elastic plate equation describing large deflections of a flexible part of the boundary. In the damped case under some conditions concerning the plate nonlinearity we prove the existence of a compact global attractor for the corresponding dynamical system and describe the situations when this attractor has a finite fractal dimension.
keywords: nonlinear plate linearized 3D Euler equations global attractor. well-posedness Fluid--structure interaction
Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory
Tomás Caraballo I. D. Chueshov Pedro Marín-Rubio José Real
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 253-270 doi: 10.3934/dcds.2007.18.253
The existence and uniqueness of solutions for a stochastic reaction-diffusion equation with infinite delay is proved. Sufficient conditions ensuring stability of the zero solution are provided and a possibility of stabilization by noise of the deterministic counterpart of the model is studied.
keywords: materials with memory stabilization. mean square exponential stability Stochastic heat equation
Pullback attractors for stochastic heat equations in materials with memory
Tomás Caraballo José Real I. D. Chueshov
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 525-539 doi: 10.3934/dcdsb.2008.9.525
We study the asymptotic behaviour of a non-autonomous stochastic reaction-diffusion equation with memory. In fact, we prove the existence of a random pullback attractor for our stochastic parabolic PDE with memory. The randomness enters in our model as an additive Hilbert valued noise. We first prove that the equation generates a random dynamical system (RDS) in an appropriate phase space. Due to the fact that the memory term takes into account the whole past history of the phenomenon, we are not able to prove compactness of the generated RDS, but its asymptotic compactness, ensuring thus the existence of the random pullback attractor.
keywords: stochastic heat equation with memory. Random pullback attractor asymptotic compactness Ornstein-Uhlenbeck process

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