10 - Partial differential equations and nonlinear semigroups

Summary

The second part of this book concentrates on the implications of Theorem 8.1 (embedding into ℝ^k for sets with finite upper box-counting dimension) for the attractors of infinite-dimensional dynamical systems.

Nonlinear semigroups and attractors

We will consider (for the most part) abstract dynamical systems defined on a real Banach space ℬ with the dynamics given by a nonlinear semigroup of solution operators, S(t) : ℬ → ℬ defined for t ≥ 0, that satisfy

1. (i) S(0) = id,

2. (ii) S(t)S(s) = S(t + s) for all t, s ≥ 0, and

3. (iii) S(t)x continuous in t and x.

Such semigroups can be generated by the solutions of partial differential equations, as we will outline in Sections 10.3 and 10.4. (At other points it will be useful to consider instead a dynamical system that arises from iterating a fixed function S : ℬ → ℬ such a map could be derived from a continuous time system by setting S = S(T) for some fixed T > 0.) An attractor for S(·) is a compact invariant set that attracts all bounded sets.

The general theory of such semigroups and their attractors is covered in detail in Chepyzhov & Vishik (2002), Chueshov (2002), Hale (1988), Robinson (2001), Sell & You (2002), and Temam (1988); we give a brief overview of the existence theory for attractors in Chapter 11, and discuss a very general method for showing that an attractor has finite upper box-counting dimension in Chapter 12.