Finite-Dimensionality of Tempered Random Uniform Attractors

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in $$L^2$$ L 2 is established by the squeezing approach and that in $$H_0^1$$ H 0 1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

The Exponential Attractor for a Class of Nonlinear Coupled Kirchhoff Equations with Strong Linear Damping

This paper investigates the dynamics for a class of nonlinear higher-order coupled Kirchhoff equations with strong linear damping. By means of the method proposed by Eden et al., the Lipschitz continuity and the discrete squeezing property of its solution semigroup are proved, and thus the existence of the exponential attractor is obtained.

Exponential attractors and inertial manifolds for a class of generalized nonlinear Kirchhoff-Sine-Gordon equation*

In this paper,we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation in n dimensional space.We first prove the squeezing property of the nonlinear semigroup associated with this equation and the existence of exponential attractors.Then using the Hadamards graph transformation method,the existence of inertial manifolds of the equation is obtained when N is sufficiently large.

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

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.

Exponential Attractor for Coupled Ginzburg-Landau Equations Describing Bose-Einstein Condensates and Nonlinear Optical Waveguides and Cavities

The existence of the exponential attractors for coupled Ginzburg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities with periodic initial boundary is obtained by showing Lipschitz continuity and the squeezing property.