Global Attractors of Evolutionary Systems

An abstract framework for studying the asymptotic behavior of a dissipative evolutionary system with respect to weak and strong topologies was introduced in Cheskidov and Foias (J Differ Equ 231:714–754, 2006) primarily to study the long-time behavior of the 3D Navier-Stokes equations (NSE) for which the existence of a semigroup of solution operators is not known. Each evolutionary system possesses a global attractor in the weak topology, but does not necessarily in the strong topology. In this paper we study the structure of a global attractor for an abstract evolutionary system, focusing on omega-limits and attracting, invariant, and quasi-invariant sets. We obtain weak and strong uniform tracking properties of omega-limits and global attractors. In addition, we discuss a trajectory attractor for an evolutionary system and derive a condition under which the convergence to the trajectory attractor is strong.     

Random attractors

In this paper, we generalize the notion of an attractor for the stochastic dynamical system introduced in [7]. We prove that the stochastic attractor satisfies most of the properties satisfied by the usual attractor in the theory of deterministic dynamical systems. We also show that our results apply to the stochastic Navier-Stokes equation, the white noise-driven Burgers equation, and a nonlinear stochastic wave equation.     

Global and Trajectory Attractors for a Nonlocal Cahn–Hilliard–Navier–Stokes System

The Cahn–Hilliard–Navier–Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier–Stokes equations suitably coupled with a nonlocal Cahn–Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn–Hilliard–Navier–Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J.M. Ball’s approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn–Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies a dissipative estimate. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force.     

Global attractor for Klein-Gordon-Schrodinger lattice system

We considered the longtime behavior of solutions of a coupled lattice dynamical system of Klein-Gordon-Schrodinger equation (KGS lattice system). We first proved the existence of a global attractor for the system considered here by introducing an equivalent norm and using "End Tails" of solutions. Then we estimated the upper bound of the Kolmogorov delta-entropy of the global attractor by applying element decomposition and the covering property of a polyhedron by balls of radii delta in the finite dimensional space. Finally, we presented an approximation to the global attractor by the global attractors of finite-dimensional ordinary differential systems.     

On Finite Fractal Dimension of the Global Attractor for the Wave Equation with Nonlinear Damping

We prove that the global attractor to a semilinear damped wave equation has finite fractal dimension provided that the damping function and the lower order nonlinearity are smooth with certain polynomial growth.     

Exponential attractors of optimal Lyapunov dimension for Navier-Stokes equations

In this paper we present a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set. The fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapunov exponents. We discuss various applications to Navier-Stokes systems.     

Trajectory and Global Attractors of the Boundary Value Problem for Autonomous Motion Equations of Viscoelastic Medium

We introduce the concept of minimal trajectory attractor generalizing the known concept of trajectory attractor of an abstract evolution equation. We obtain several results on existence and properties of minimal trajectory and global attractors without assumptions of any invariance of the trajectory space of an equation. With the help of these results we prove existence of minimal trajectory and global attractors for weak solutions of the boundary value problem for autonomous motion equations of an incompressible viscoelastic medium with the Jeffreys constitutive law. The work was partially supported by grants 04-01-00081 of Russian Foundation of Basic Research, VZ-010-0 of the Ministry of Education and Science of Russia and CRDF and MK- 3650.2005.1 of President of Russian Federation.     

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

Finite Dimensional Attractor for a One-Phase Stefan Problem with Kinetics

For a one-phase free-boundary problem with kinetics, which is known to generate a rich dynamics, we study evolution of the infinitesimal volume along the trajectories in the attractor. We demonstrate that for sufficiently large m that is defined solely by the properties of the kinetics function the m-dimensional volume decays exponentially. This property combined with the uniform differentiability of the semigroup leads to the conclusion that the Hausdorff dimension of the attractor is finite.     

The attractor of a Navier-Stokes system in an unbounded channel-like domain

The Navier-Stokes system describes a flow of a fluid in an unbounded planar channel-like domain. It is proved that in the case when an external force decays at infinity, the semigroup generated by this system has a global attractor and its Hausdorff dimension is finite. Estimates in weighted Sobolev spaces are used as a main tool. Asymptotics, as the distance from the origin in the plane tends to infinity, of functions on the attactor is found. This asymptotics show that all dynamics on the attractor decays at infinity and the turbulence generated by the force does not propagate to infinity.This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

Attactors of dissipative soliton equation

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Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison

We consider two magnetohydrodynamic-α (MHDα) models with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). More precisely, we consider the Navier–Stokes-α-MHD and the Modified Leray-α-MHD models. Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations. In both cases, the global existence of the solution and of a global attractor can be shown. We provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the long-time dynamics of the involved system and gives information about the numerical stability of the model. We get the same bound that holds for the Simplified Bardina-MHD model, considered in a previous paper (this result provides, in some sense, an intermediate bound between the number of degrees of freedom for the Simplified Bardina model and the Navier–Stokes-α equation in the nonmagnetic case). However, the Navier–Stokes-α-MHD system is preferable since, in the ideal case, it conserves more quadratic invariants derived from the standard MHD model.     

Global attractor for Klein-Gordon-Schroedinger lattice system

We considered the longtime behavior of solutions of a coupled lattice dynamical system of Klein-Gordon-Schroedinger equation （KGS lattice system）. We first proved the existence of a global attractor for the system considered here by introducing an equivalent norm and using ＂End Tails＂ of solutions. Then we estimated the upper bound of the Kolmogorov delta-entropy of the global attractor by applying element decomposition and the covering property of a polyhedron by balls of radii delta in the finite dimensional space. Finally, we presented an approximation to the global attractor by the global attractors of finite-dimensional ordinary differential systems.     

Global attractors for the three-dimensional Navier-Stokes equations

In this paper we show that the weak solutions of the Navier-Stokes equations on any bounded, smooth three-dimensional domain have a global attractor for any positive value of the viscosity. The proof of this result, which bypasses the two issues of the possible nonuniqueness of the weak solutions and the possible lack of global regularity of the strong solutions, is based on a new point of view for the construction of the semiflow generated by these equations. We also show that, under added assumptions, this global attractor consists entirely of strong solutions.     

Local and Global Lyapunov exponents

Various properties of Local and Global Lyapunov exponents are related by redefining them as the spectral radii of some positive operators on a space of continuous functions and utilizing the theory developed by Choquet and Foias. These results are then applied to the problem of estimating the Hausdorff dimension of the global attractor and the existence of a critical trajectory, along which the Lyapunov dimension is majorized, is established. Using this new estimate, the existing dimension estimate for the global attractor of the Lorenz system is improved. Along the way a simple relation between topological entropy and the fractal dimension is obtained.     

Local Exact Controllability for the One-Dimensional Compressible Navier–Stokes Equation

In this paper we deal with the isentropic (compressible) Navier-Stokes equation in one space dimension and we adress the problem of the boundary controllability for this system. We prove that we can drive initial conditions which are sufficiently close to some constant states to those constant states. This is done under some natural hypotheses on the time of control and on the regularity on the initial conditions.     

Regularity and finite dimensionality of attractor for plate equation on R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

This paper addresses the regularity and finite dimensionality of the global attractor for the plate equation on the unbounded domain. The existence of the attractor in the phase space has been established in an earlier work of the author. It is shown that the attractor is actually a bounded set of the phase space and has finite fractal dimensionality.     

Singular limit of equations for linear viscoelastic fluids with periodic boundary conditions

We consider a one-parameter family of problems, governing, for any fixed parameter, the motion of a linear viscoelastic fluid in a two-dimensional domain with periodic boundary conditions. The asymptotic behavior of each problem is analyzed, by proving the existence of the global attractor. Moreover, letting the parameter go to zero, since the memory effect disappears, we obtain a limiting problem, given by the Navier-Stokes equations. For any fixed parameter, we construct an exponential attractor. The resulting family is robust, meaning that these exponential attractors converge, in an appropriate sense, to an exponential attractor of the limiting problem.     

Pseudotrajectories generated by a discretization of a parabolic equation

A semiimplicit discretization of a parabolic equation is considered. The resulting diffepmorphism is shown to be generically Morse-Smale. Uniform bounds for the dimension of its attractor are given and numerical trajectories—including round-off errors—are shown to approximate the attractor.     

Positive Solutions to Singular Second and Third Order Differential Equations for Quantum Fluids

We analyze a quantum trajectory model given by a steady-state hydrodynamic system for quantum fluids with positive constant temperature in bounded domains for arbitrary large data. The momentum equation can be written as a dispersive third-order equation for the particle density where viscous effects are incorporated. The phenomena that admit positivity of the solutions are studied. The cases, one space dimensional dispersive or non-dispersive, viscous or non-viscous, are thoroughly analyzed with respect to positivity and existence or non-existence of solutions, all depending on the constitutive relation for the pressure law. We distinguish between isothermal (linear) and isentropic (power law) pressure functions of the density. It is proved that in the dispersive, non-viscous model, a classical positive solution only exists for “small” (positive) particle current densities, both for the isentropic and isothermal case. Uniqueness is also shown in the isentropic subsonic case, when the pressure law is strictly convex. However, we prove that no weak isentropic solution can exist for “large” current densities. The dispersive, viscous problem admits a classical positive solution for all current densities, both for the isentropic and isothermal case, with an “ultra-diffusion” condition. The proofs are based on a reformulation of the equations as a singular elliptic second-order problem and on a variant of the Stampacchia truncation technique. Some of the results are extended to general third-order equations in any space dimension. Accepted July 1, 2000?Published online February 14, 2001