Kolmogorov $\varepsilon$-entropy estimates for the~uniform attractors of non-autonomous reaction-diffusion systems

M. I. Vishik ; V. V. Chepyzhov

Sbornik. Mathematics, Tome 189 (1998) no. 2, pp. 235-263

Résumé

The Kolmogorov $\varepsilon$-entropy of the uniform attractor $\mathscr A$ of a family of non-autonomous reaction-diffusion systems with external forces $g(x,t)$ is studied. The external forces $g(x,t)$ are assumed to belong to some subset $\sigma$ of $C({\mathbb R};H)$, where $H=(L_2(\Omega ))^N$, that is invariant under the group of $t$-translations. Furthermore, $\sigma$ is compact in $C({\mathbb R};H)$. An estimate for the $\varepsilon$-entropy of the uniform attractor $\mathscr A$ is given in terms of the $\varepsilon _1=\varepsilon _1(\varepsilon )$-entropy of the compact subset $\sigma_l$ of $C([0,l];H)$ consisting of the restrictions of the external forces $g(x,t)\in \sigma$ to the interval $[0,l]$, $l=l(\varepsilon )$ ($\varepsilon _1(\varepsilon )\sim \mu \varepsilon $, $l(\varepsilon )\sim \tau \log _2(1/\varepsilon )$). This general estimate is illustrated by several examples from different fields of mathematical physics and information theory.