scholarly journals Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with 'fading memory'

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Manil T. Mohan ◽  
Keyword(s):  
Three Dimensional ◽  
Global Attractors ◽  
Fading Memory ◽  
Exponential Attractors ◽  
Determining Modes

scholarly journals Quasi-stability and continuity of attractors for nonlinear system of wave equations

2021 ◽  
Vol 8 (1) ◽  
pp. 27-45
Author(s):  
M. M. Freitas ◽  
M. J. Dos Santos ◽  
A. J. A. Ramos ◽  
M. S. Vinhote ◽  
M. L. Santos
Keyword(s):  
Wave Equations ◽  
Coupled System ◽  
Global Attractors ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Small Perturbations ◽  
Recent Approach ◽  
Nonlinear Coupled System ◽  
Long Time ◽  
External Forces

Abstract In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

scholarly journals Parametrization of global attractors, experimental observations, and turbulence

2007 ◽  
Vol 578 ◽  
pp. 495-507 ◽  
Author(s):  
JAMES C. ROBINSON
Keyword(s):  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Global Attractors ◽  
Navier Stokes ◽  
Minimum Length ◽  
Abstract Theory ◽  
Rigorous Results ◽  
Spatially Extended ◽  
Minimum Length Scale

This paper is concerned with rigorous results in the theory of turbulence and fluid flow. While derived from the abstract theory of attractors in infinite-dimensional dynamical systems, they shed some light on the conventional heuristic theories of turbulence, and can be used to justify a well-known experimental method.Two results are discussed here in detail, both based on parametrization of the attractor. The first shows that any two fluid flows can be distinguished by a sufficient number of point observations of the velocity. This allows one to connect rigorously the dimension of the attractor with the Landau–Lifschitz ‘number of degrees of freedom’, and hence to obtain estimates on the ‘minimum length scale of the flow’ using bounds on this dimension. While for two-dimensional flows the rigorous estimate agrees with the heuristic approach, there is still a gap between rigorous results in the three-dimensional case and the Kolmogorov theory.Secondly, the problem of using experiments to reconstruct the dynamics of a flow is considered. The standard way of doing this is to take a number of repeated observations, and appeal to the Takens time-delay embedding theorem to guarantee that one can indeed follow the dynamics ‘faithfully’. However, this result relies on restrictive conditions that do not hold for spatially extended systems: an extension is given here that validates this important experimental technique for use in the study of turbulence.Although the abstract results underlying this paper have been presented elsewhere, making them specific to the Navier–Stokes equations provides answers to problems particular to fluid dynamics, and motivates further questions that would not arise from within the abstract theory itself.

A squeezing property and its applications to a description of long-time behaviour in the three-dimensional viscous primitive equations

2014 ◽  
Vol 144 (4) ◽  
pp. 711-729 ◽  
Author(s):  
Igor Chueshov
Keyword(s):  
Three Dimensional ◽  
Ocean Dynamics ◽  
Primitive Equations ◽  
Time Behaviour ◽  
Time Dynamics ◽  
Squeezing Property ◽  
Long Time Behaviour ◽  
New Information ◽  
Determining Modes ◽  
Long Time

We consider the three-dimensional viscous primitive equations with periodic boundary conditions. These equations arise in the study of ocean dynamics and generate a dynamical system in a Sobolev H1-type space. Our main result establishes the so-called squeezing property in the Ladyzhenskaya form for this system. As a consequence of this property we prove the finiteness of the fractal dimension of the corresponding global attractor, the existence of a finite number of determining modes and the ergodicity of a related random kick model. All these results provide new information concerning the long-time dynamics of oceanic motion.

scholarly journals The global attractors and exponential attractors for a class of nonlinear damping Kirchhoff equation

2016 ◽  
Vol 12 (10) ◽  
pp. 6686-6704
Author(s):  
Huixian Zhu ◽  
Chengfei Ai ◽  
Guoguang Lin
Keyword(s):  
Initial Boundary ◽  
Global Attractors ◽  
Nonlinear Damping ◽  
Time Behavior ◽  
Semi Group ◽  
Priori Estimate ◽  
Exponential Attractors ◽  
Long Time ◽  
Initial Boundary Value ◽  
The Galerkin Method

This paper consider the long time behavior of a class of nonlinear damped Kirchhoff equation    2 tt 1 t t u u u u u f x          . Study the attractor problem with initial boundary value conditions, then using priori estimate and the Galerkin method prove existence and uniqueness of solution, we obtain to the existence of the global attractors. The squeezing property of the nonlinear semi-group associated with this equation and the existence of exponential attractors are also proved.

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