Abstract
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 is established by the squeezing approach and that in 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.
Highlights
Attractor theory is known as a useful tool in the study of infinite-dimensional dynamical systems, especially in numerical simulations and computations
If a system has an attractor, any solution trajectory of the system can be tracked by trajectories within the attractor, while, in the if the attractor has finite dimension, finite degrees of freedom are expected to fully determine the asymptotic behavior of the system, though the phase space of the system is infinite dimensional. This is known as a finite-dimensional reduction of infinite-dimensional dynamical systems (Temam 1997; Robinson 2011)
The non-autonomous feature of the system prevents one to learn from these pullback attractors about the forward dynamics of the underlying system, and this is the motivation of our previous work (Cui and Langa 2017), where a random uniform attractor was developed, which provides a certain description of forward dynamics of the system by the property of uniformly forward attracting in probability
Summary
Attractor theory is known as a useful tool in the study of infinite-dimensional dynamical systems, especially in numerical simulations and computations. Since the study of uniform attractors is usually based on a symbol space which contains auxiliary elements that could not belong to the original system, a uniform attractor is more often infinite dimensional It has been an untouched problem for almost twenty years that. By a smoothing property of the underlying system, we established criteria for a uniform attractor to have finite fractal dimension, and the upper bound consists of two parts: the fractal dimension of the symbol space plus an auxiliary number arising from the smoothing property This structure of the upper bound agrees with the result (Chepyzhov and Vishik 2002, Theorem IX 2.1) of Chepyzhov and Vishik established by studying the quasi-differentials of the system.
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