Kolmogorov ε-Entropy in the Problems on Global Attractors for Evolution Equations of Mathematical Physics

We study the Kolmogorov ε-entropy and the fractal dimension of global attractors for autonomous and nonautonomous equations of mathematical physics. We prove upper estimates for the ε-entropy and fractal dimension of the global attractors of nonlinear dissipative wave equations. Andrey Nikolaevich Kolmogorov discovered applications of notions of information theory in the theory of dynamical systems. In particular, he introduced the notion of ε-entropy Hε(X) of a compact set X in a Banach space E. The well-known paper of Kolmogorov and V.M. Tikhomirov [1] contains many important estimates from above and from below for the ε-entropy Hε(X) of a number of particular function sets X. For example, in the paper, the ε-entropy is studied for the set of real functions {u(t), t ∈ R} that have bounded spectrum, and a variant of the Kotelnikov theorem is proved (see also [2]). In the last decades, global attractors A were intensively investigated for basic evolution equations of mathematical physics, for which the initial Cauchy problem is studied deep enough. Recall that a global attractor A is a compact set in the corresponding Banach or Hilbert space that obeys the invariance property with respect to the corresponding dynamical system and that attracts bounded sets of trajectories as time t→ +∞. For certainty and brevity, in this paper we study the Kolmogorov ε-entropy and fractal dimension of a global attractor A of the dissipative wave equation in a bounded domain Ω b Rn. We consider in more detail the case of the sine-Gordon equation, where the global attractor admits a simple structure. For the case of an autonomous hyperbolic equation, where all coefficients and the exiting force of the equation do not depend on time, we present in Section 2 an upper estimate for the ε-entropy Hε(A) of its global attractor A and give an estimate from above for the fractal dimension dF (A) of this attractor. Before this, we formulate the general theorem on the estimation of the ε-entropy Hε(X) of an invariant set X of an abstract autonomous dynamical system. Section 3 is devoted to the construction of the global attractor A for the nonautonomous wave equation as well as for an abstract nonautonomous evolution equation. In Section 4, we prove an upper estimate for the ε-entropy Hε(A) of the global attractor A of the nonautonomous wave equation with, for example, exiting force g(x, t), x ∈ Ω, that depends on time t. We also present in this section some general facts concerning abstract nonautonomous dynamical systems. In the particular case where the exiting force g(x, t) of the wave equation is an almost periodic function of t, the ε-entropy Hε(A) of the global attractor A for 0 < ε ≤ ε0 does not exceed the sum of three terms: 1 Supported in part by the Russian Foundation for Fundamental Research, project no. 02-01-00227; CRDF, grant no. RM1-2343-MO-02; and INTAS, grant no. 00-899. 0032-9460/03/3901-0002 $25.00 c © 2003 MAIK “Nauka/Interperiodica” KOLMOGOROV ε-ENTROPY OF GLOBAL ATTRACTORS 3 first, Hβ(H(g)), where H(g) is the hull of the function g(x, t) in the space Cb(R;L2(Ω)), β > 0, and we give an explicit expression for the value β = β(ε) (see Section 4); second, D log2 (1/ε), this term is analogous to those encountered in the upper estimate of the ε-entropy of the attractor A of an autonomous hyperbolic equation; third, Hε0(A), where ε0 is fixed. It should be noted that the fractal dimension dF (A) of the global attractor A in the nonautonomous case can be infinite. However, if the function g(x, t) is quasiperiodic in t, that is, g(x, t) = G(x, α1t, . . . , αkt), where G(x, ω1, . . . , ωk) is a 2π-periodic function in each variable ωi, i = 1, . . . , k, then the fractal dimension dF (A) is finite. In this case, dF (A) ≤ dF (H(g)) + D ≤ k + D and the ε-entropy Hε(A) . k log2 ( 1 ε ) + D log2 ( 1 ε ) . Here k is the number of rationally independent frequencies {αi}, i = 1, . . . , k, of the quasiperiodic function g(x, t). In particular, for k = 0, we get an estimate for the autonomous equation. In conclusion, note that the Kolmogorov ε-entropy Hε(A) of the global attractor A is always finite because the set A is compact in the corresponding function space. The behavior of the quantity Hε(A) as a function of ε as ε→ 0+ describes the complexity of the global attractor A of the dynamical system under consideration. 1. ε-ENTROPY AND FRACTAL DIMENSION OF COMPACT SETS Let us formulate the definition of the Kolmogorov ε-entropy of a compact set X in a Banach space E. Denote by Nε(X,E) = Nε(X) the minimum number of open balls in E of radius ε which is necessary to cover X: