scholarly journals Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid

2009 ◽  
Vol 247 (8) ◽  
pp. 2331-2363 ◽  
Author(s):  
Caidi Zhao ◽  
Yongsheng Li ◽  
Shengfan Zhou
Keyword(s):  
Global Attractor ◽  
Newtonian Fluid ◽  
Upper Semicontinuity ◽  
Trajectory Attractor

scholarly journals Existence and approximation of attractors for nonlinear coupled lattice wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lianbing She ◽  
Mirelson M. Freitas ◽  
Mauricio S. Vinhote ◽  
Renhai Wang
Keyword(s):  
Asymptotic Behavior ◽  
Global Attractor ◽  
Wave Equations ◽  
Upper Semicontinuity ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Compactness ◽  
Finite Dimensional ◽  
Lattice Wave ◽  
Infinite Lattices

<p style='text-indent:20px;'>This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in <inline-formula><tex-math id="M1">\begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document}</tex-math></inline-formula>. We then prove that the solution semigroup has a unique global attractor in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula>. We finally prove that this attractor can be approximated in terms of upper semicontinuity of <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula> by a finite-dimensional global attractor of a <inline-formula><tex-math id="M4">\begin{document}$ 2(2n+1) $\end{document}</tex-math></inline-formula>-dimensional truncation system as <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.</p>

UPPER SEMICONTINUITY OF ATTRACTORS FOR THE KLEIN–GORDON–SCHRÖDINGER EQUATION

2005 ◽  
Vol 15 (01) ◽  
pp. 157-168 ◽  
Author(s):  
KENING LU ◽  
BIXIANG WANG
Keyword(s):  
Schrödinger Equation ◽  
Global Attractor ◽  
Schrodinger Equation ◽  
Upper Semicontinuity ◽  
Global Attractors ◽  
Klein Gordon

In this paper, we consider the Klein–Gordon–Schröodinger equation defined on Rn (n ≤ 3) and Ωm = {x ∈ Rn : |x| ≤ m}. Let [Formula: see text] and [Formula: see text] be the global attractors of the equation corresponding to Rn and Ωm, respectively. Then we prove that for any neighborhood U of [Formula: see text], the global attractor [Formula: see text] enters U when m is large enough.

scholarly journals Upper Semicontinuity of Trajectory Attractors for 3D Incompressible Navier–Stokes Equation

2019 ◽  
Author(s):  
Yuming Qin ◽  
Xiuqing Wang
Keyword(s):  
Stokes Equation ◽  
Upper Semicontinuity ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Trajectory Attractor ◽  
Incompressible Navier Stokes Equation ◽  
Trajectory Attractors

Abstract In this paper, we first establish the existence of a trajectory attractor for the Navier–Stokes–Voight (NSV) equation and then prove upper semicontinuity of trajectory attractors of 3D incompressible Navier–Stokes equation when 3D NSV equation is considered as a perturbative equation of 3D incompressible Navier–Stokes equation.

scholarly journals Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation

2005 ◽  
Vol 2005 (6) ◽  
pp. 655-671 ◽  
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Global Attractor ◽  
Fourth Order ◽  
Boussinesq Equation ◽  
Upper Semicontinuity ◽  
Time Variable ◽  
Finite Dimensional ◽  
Lattice Dynamical System ◽  
Second Order In Time

We will study the lattice dynamical system of a nonlinear Boussinesq equation. Our objective is to explore the existence of the global attractor for the solution semiflow of the introduced lattice system and to investigate its upper semicontinuity with respect to a sequence of finite-dimensional approximate systems. As far as we are aware, our result here is the first concerning the lattice dynamical system corresponding to a differential equation of second order in time variable and fourth order in spatial variable with nonlinearity involving the gradients.

scholarly journals DYNAMICS OF A REACTION–DIFFUSION EQUATION WITH A DISCONTINUOUS NONLINEARITY

2006 ◽  
Vol 16 (10) ◽  
pp. 2965-2984 ◽  
Author(s):  
JOSÉ M. ARRIETA ◽  
ANÍBAL RODRÍGUEZ-BERNAL ◽  
JOSÉ VALERO
Keyword(s):  
Nonlinear Dynamics ◽  
Diffusion Equation ◽  
Fixed Points ◽  
Global Attractor ◽  
Nonlinear Term ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Discontinuous Nonlinearity ◽  
Smooth Approximations

We study the nonlinear dynamics of a reaction–diffusion equation where the nonlinearity presents a discontinuity. We prove the upper semicontinuity of solutions and the global attractor with respect to smooth approximations of the nonlinear term. We also give a complete description of the set of fixed points and study their stability. Finally, we analyze the existence of heteroclinic connections between the fixed points, obtaining information on the fine structure of the global attractor.

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