Smoothing and finite-dimensionality of uniform attractors in Banach spaces

• DOI: 10.1016/j.jde.2021.03.013
• Publication Date: 2021-03-18T03:42:19Z
• Authors: Hongyong Cui; Alexandre N. Carv...; Arthur C. Cunha; José A. Langa
• Keywords: Dimension function; Smoothing

Abstract The aim of this paper is to find an upper bound for the fractal dimension of uniform attractors in Banach spaces. The main technique we employ is essentially based on a compact embedding of some auxiliary Banach space into the phase space and a corresponding smoothing effect between these spaces. Our bounds on the fractal dimension of uniform attractors are given in terms of the dimension of the symbol space and the Kolmogorov entropy number of the embedding. In addition, a dynamical analysis on the symbol space is also given, showing that the finite-dimensionality of the hull of a time-dependent function is fully determined by the tails of the function, which allows us to consider more general non-autonomous terms than quasi-periodic functions. As applications, we show that the uniform attractor of the 2D Navier-Stokes equation is finite-dimensional in H and in V, and that of a reaction-diffusion equation is finite-dimensional in L 2 and in L p , with p > 2 .