Random attractors for stochastic reaction–diffusion equations on unbounded domains

The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations

We consider a model for the flow of a mixture of two homogeneous and incompressible fluids in a two-dimensional bounded domain. The model consists of a Navier–Stokes equation governing the fluid velocity coupled with a convective Cahn–Hilliard equation for the relative density of atoms of one of the fluids. Endowing the system with suitable boundary and initial conditions, we analyze the asymptotic behavior of its solutions. First, we prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses the global attractor A . Then we establish the existence of an exponential attractors E . Thus A has finite fractal dimension. This dimension is then estimated from above in terms of the physical parameters. Moreover, assuming the potential to be real analytic and in absence of volume forces, we demonstrate that each trajectory converges to a single equilibrium. We also obtain a convergence rate estimate in the phase-space metric.

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Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D

This paper presents an abstract framework for the optimization of actuating and sensing devices in distributed parameter systems. The optimization is performed on the repositioning of these devices throughout the spatial domain. It is assumed that a network of mobile sensing and actuating devices is available to obtain measurements from the spatially distributed process and dispense control signals to the spatially distributed process. By taking advantage of the properties that the spatial operator that governs the process dynamics possesses, namely symmetry, coercivity and boundedness, a scheme for the guidance of a mobile actuator-plus-sensor network is developed and used for the performance enhancement of the spatial process. The class of systems is governed by diffusion PDEs equipped with actuators having a boxcar spatial distribution and a collocated sensing device that provides the spatially averaged state measurement over the range of the actuating device. Using Lyapunov stability arguments, a stable guidance scheme is provided for each of the mobile agents. Due to the specific structure of the closed loop system operator with time-varying input and output operators, the same guidance schemes are applicable to the dual problem of mobile sensors employed for enhancing the state estimation problem. Both a centralized state estimator with mobile sensors and a network of consensus distributed estimators are considered, since both filters can be shown to result in the specific algebraic structure of a symmetric spatial operator with collocated input and output operators. Extensive numerical simulations for a 1-D diffusion equation with two actuator/sensor agents, a 2-D diffusion equation with one collocated actuator/sensor agent plus a centralized filter for a 1-D diffusion equation are included to verify the effectiveness of a such a mobile actuator-plus-sensor network in suppressing the effects of spatially varying disturbances and enhancing the system's performance.

Guidance of Mobile Actuator-Plus-Sensor Networks for Improved Control and Estimation of Distributed Parameter Systems

The
existence and structure of uniform attractors in $V$ is proved for
nonautonomous 2D Navier-stokes equations on bounded domain with a
new class of external forces, termed normal in
$L_{l o c}^2(\mathbb R; H)$ (see Definition 3.1), which are
translation bounded but not translation compact in
$L_{l o c}^2(\mathbb R; H)$. To this end, some abstract
results are established. First, a characterization on the
existence of uniform attractor for a family of processes is
presented by the concept of measure of noncompactness as well as a
method to verify it. Then, the structure of the uniform attractor
is obtained by constructing skew product flow on the extended
phase space with weak topology. Finally, the uniform attractor of
a process is identified with that of a family of processes with
symbols in the closure of the translation family of the original
symbol in a Banach space with weak topology.

Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces

Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge Texts in Applied Mathematics

JC Robinson

Papers

Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge Texts in Applied Mathematics