Cooperative random and stochastic differential equations
Ludwig Arnold Igor Chueshov
Discrete & Continuous Dynamical Systems - A 2001, 7(1): 1-33 doi: 10.3934/dcds.2001.7.1
This is a systematic study of order-preserving (or monotone) random dynamical systems which are generated by cooperative random or stochastic differential equations. Our main results concern the long-term behavior of these systems, in particular the existence of equilibria and attractors and a limit set trichotomy theorem. Several applications (models of the control of the protein synthesis in a cell, of gonorrhea infection and of symbiotic interaction in a random environment) are treated in detail.
keywords: order-preserving or monotone random dynamical system random attractor long-term behavior Cooperative differential equation random equilibrium limit set trichotomy.
Long-time dynamics in plate models with strong nonlinear damping
Igor Chueshov Stanislav Kolbasin
Communications on Pure & Applied Analysis 2012, 11(2): 659-674 doi: 10.3934/cpaa.2012.11.659
We study long-time dynamics of a class of abstract second order in time evolution equations in a Hilbert space with the damping term depending both on displacement and velocity. This damping represents the nonlinear strong dissipation phenomenon perturbed with relatively compact terms. Our main result states the existence of a compact finite dimensional attractor. We study properties of this attractor. We also establish the existence of a fractal exponential attractor and give the conditions that guarantee the existence of a finite number of determining functionals. In the case when the set of equilibria is finite and hyperbolic we show that every trajectory is attracted by some equilibrium with exponential rate. Our arguments involve a recently developed method based on the "compensated" compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, Karman and Berger plate models with different types of boundary conditions and strong damping terms. Our results can be also applied to the nonlinear wave equations.
keywords: global attractor Nonlinear plate models state-dependent damping
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.
Stochastic dynamics in a fluid--plate interaction model with the only longitudinal deformations of the plate
Igor Chueshov Björn Schmalfuß
Discrete & Continuous Dynamical Systems - B 2015, 20(3): 833-852 doi: 10.3934/dcdsb.2015.20.833
We consider a stochastically perturbed coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a random pullback attractor of finite fractal dimension. Our argument is based on some modification of the method of quasi-stability estimates recently developed for deterministic systems.
keywords: quasi-stability method. random pullback attractor finite fractal dimension stochastic forcing Fluid-plate interaction
On interaction of circular cylindrical shells with a Poiseuille type flow
Igor Chueshov Tamara Fastovska
Evolution Equations & Control Theory 2016, 5(4): 605-629 doi: 10.3934/eect.2016021
We study dynamics of a coupled system consisting of the 3D Navier--Stokes equations which is linearized near a certain Poiseuille type flow between two unbounded circular cylinders and nonlinear elasticity equations for the transversal displacements of the bounding cylindrical shells. We show that this problem generates an evolution semigroup $S_t$ possessing a compact finite-dimensional global attractor.
keywords: Fluid--structure interaction nonlinear circular shell attractor. Poiseulle flow linearized 3D Navier--Stokes equations
Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay
Igor Chueshov Alexander V. Rezounenko
Communications on Pure & Applied Analysis 2015, 14(5): 1685-1704 doi: 10.3934/cpaa.2015.14.1685
We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz in time. This allows us to show that the model considered generates an evolution operator semigroup $S_t$ on a certain space of Lipschitz type functions over delay time interval. The operators $S_t$ are closed for all $t\ge 0$ and continuous for $t$ large enough. Our main result shows that the semigroup $S_t$ possesses compact global and exponential attractors of finite fractal dimension. Our argument is based on the recently developed method of quasi-stability estimates and involves some extension of the theory of global attractors for the case of closed evolutions.
keywords: state-dependent delay exponential attractor. finite-dimension global attractor Parabolic evolution equations
Invariance and monotonicity for stochastic delay differential equations
Igor Chueshov Michael Scheutzow
Discrete & Continuous Dynamical Systems - B 2013, 18(6): 1533-1554 doi: 10.3934/dcdsb.2013.18.1533
We study invariance and monotonicity properties of Kunita-type sto-chastic differential equations in $\mathbb{R}^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of $\mathbb{R}^d$. Then we present a comparison principle and show that under appropriate conditions the stochastic delay system considered generates a monotone (order-preserving) random dynamical system. Several applications are considered.
keywords: random attractor. Stochastic delay/functional differential equation invariance stochastic flow random dynamical system monotonicity
Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models
Igor Chueshov Irena Lasiecka
Discrete & Continuous Dynamical Systems - A 2006, 15(3): 777-809 doi: 10.3934/dcds.2006.15.777
We study dynamics of a class of nonlinear Kirchhoff-Boussinesq plate models. The main results of the paper are: (i) existence and uniqueness of weak (finite energy) solutions, (ii) existence of weakly compact attractors.
keywords: 2D Boussinesq models weak well-posedness global attractors.
Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions
Igor Chueshov Björn Schmalfuss
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 315-338 doi: 10.3934/dcds.2007.18.315
We consider non-linear parabolic stochastic partial differential equations with dynamical boundary conditions and with a noise which acts in the domain but also on the boundary and is presented by the temporal generalized derivative of an infinite dimensional Wiener process. We prove that solutions to this stochastic partial differential equation generate a random dynamical system. Under additional conditions we show that this system is monotone. Our main result states the existence of a compact global (pullback) attractor.
keywords: stochastic PDE monotonicity. random dynamical systems dynamical boundary conditions pullback attractor
Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations
Igor Chueshov Alexey Shcherbina
Evolution Equations & Control Theory 2012, 1(1): 57-80 doi: 10.3934/eect.2012.1.57
We deal with an initial boundary value problem for the Schrödinger-Boussinesq system arising in plasma physics in two-dimensional domains. We prove the global Hadamard well-posedness of this problem (with respect to the topology which is weaker than topology associated with the standard variational (weak) solutions) and study properties of the solutions. In the dissipative case the existence of a global attractor is established.
keywords: 2D Schrödinger-Boussinesq models global attractors. weak and strong solutions

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