Appendix - Non-autonomous dynamical systems and their attractors

Summary

In the Appendix we consider a new approach to the investigation of attractors for nonlinear non-autonomous partial differential equations. The Appendix is organised as follows. We start in §A1 with the definition of a process {U(t,) | t ≥, ∈ } acting on a Banach space E. §A1 contains also some examples of non-autonomous equations and systems of mathematical physics, which generate processes.

In §A2 we introduce the major concepts of absorbing set, attracting set and attractor for a given process {U(t,)}. This section also deals with a family of processes {U_f(t,) | t ≥, ∈ } depending on some functional parameter f called the time symbol of a process. A special translation property holds for such a family of processes. It is shown that the study of a uniform attractor of such a family of processes can be reduced to the investigation of some semigroup {S(t} acting on some expanded phase space. We prove the main theorem about the existence of a uniform attractor of a family of processes with almost periodic time symbols.

§A3 is aimed at applying the general results of §A2 to the equations and systems we have considered in §A1. Using the main theorem we establish the existence of a uniform attractor of a family of processes generated by the two-dimensional Navier-Stokes equations with almost periodic exterior forces.