Upper estimates on Hausdorff and fractal dimensions of global attractors for the 2D Navier–Stokes–Voight equations with a distributed delay

2019 ◽  
Vol 111 (3-4) ◽  
pp. 179-199
Author(s):  
Yuming Qin ◽  
Keqin Su
Keyword(s):  
Distributed Delay ◽  
Fractal Dimensions ◽  
Global Attractors ◽  
Navier Stokes ◽  
Hausdorff And Fractal Dimensions

scholarly journals The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation with nonlinear strongly damped terms

2016 ◽  
Vol 12 (10) ◽  
pp. 6658-6673
Author(s):  
Wei Wang ◽  
Ling Chen ◽  
Guoguang Lin
Keyword(s):  
Fractal Dimensions ◽  
Type Equation ◽  
Initial Boundary ◽  
Global Attractors ◽  
Time Behavior ◽  
Initial Boundary Value Problems ◽  
Long Time ◽  
Uniqueness Of The Solution ◽  
Hausdorff And Fractal Dimensions ◽  
Initial Boundary Value

In this paper ,we study the long time behavior of solution to the initial boundary value problems for higher -orderkirchhoff-type equation with nonlinear strongly dissipation:At first ,we prove the existence and uniqueness of the solution by priori estimate and Galerkin methodthen we establish the existence of global attractors ,at last,we consider that estimation of upper bounds of Hausdorff and fractal dimensions for the global attractors are obtain.

The Estimation of the Hausdorff and Fractal Dimensions of the Global Attractor for 2D Autonomous g-Navier-Stokes Equations

2014 ◽  
Vol 511-512 ◽  
pp. 1235-1238
Author(s):  
Jin Ping Jiang ◽  
Xiao Xia Wang
Keyword(s):  
Sobolev Space ◽  
Global Attractor ◽  
Stokes Equations ◽  
Weighted Sobolev Space ◽  
Unbounded Domains ◽  
Fractal Dimensions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Forcing Term ◽  
Hausdorff And Fractal Dimensions

In this paper, by using the energyequation method, the 2D g-Navier-Stokes equations with linear dampness on some unbounded domains wereinvestigated without the restriction of the forcing term belongingto some weighted Sobolev space. Moreover,the estimation of theHausdorff and Fractal dimensions of such attractors were alsoobtained.

scholarly journals Topological dimensions of global attractors for semilinear PDE's with delays

1991 ◽  
Vol 43 (3) ◽  
pp. 407-422 ◽  
Author(s):  
Joseph W.-H. So ◽  
Jianhong Wu
Keyword(s):  
Dynamical Systems ◽  
Product Space ◽  
Nonlinear Dynamical Systems ◽  
Fractal Dimensions ◽  
Global Attractors ◽  
Nonlinear Semigroup ◽  
Nonlinear Dynamical ◽  
Hausdorff And Fractal Dimensions ◽  
Upper Estimate ◽  
Equations With Delay

An estimate is obtained on the Hausdorff and fractal dimensions of global attractors of semilinear partial differential equations with delay: ẋ(t) = Ax(t) + f(xt). The method employed is to associate such an equation with a nonlinear semigroup on a product space and then appeal to the upper estimate due to Constantin, Foias and Teman on topological dimensions of global attractors for general nonlinear dynamical systems.

scholarly journals The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation*

2016 ◽  
Vol 12 (9) ◽  
pp. 6608-6621
Author(s):  
Ling Chen ◽  
Wei Wang ◽  
Guoguang Lin
Keyword(s):  
Existence And Uniqueness ◽  
Finite Dimension ◽  
Fractal Dimensions ◽  
Type Equation ◽  
Global Attractors ◽  
Higher Order ◽  
Well Posedness ◽  
Priori Estimate ◽  
Uniqueness Of The Solution ◽  
Hausdorff And Fractal Dimensions

We investigate the global well-posedness and the longtime dynamics of solutions for the higher-order Kirchhoff-typeequation with nonlinear strongly dissipation:2( ) ( )m mt t tu    u    D u  ( ) ( ) ( )m  u  g u  f x . Under of the properassume, the main results are that existence and uniqueness of the solution is proved by using priori estimate and Galerkinmethod, the existence of the global attractor with finite-dimension, and estimation Hausdorff and fractal dimensions of theglobal attractor.

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.

Sign in / Sign up

Export Citation Format

Share Document