# zbMATH — the first resource for mathematics

Attractors of non-autonomous dynamical systems and their dimension. (English) Zbl 0838.58021
This extensive and important paper deals with the non-autonomous evolution equations of the form: (1) $$\partial_t u= A_{\sigma(t)}(u)$$, $$t\in \mathbb{R}$$, where for any fixed $$t\in \mathbb{R}$$, $$A_{\sigma(t)}(u)$$ is a nonlinear operator acting from a Banach space $$E_1$$ into a Banach space $$E_0$$. Usually the space $$E_1$$ is dense in $$E_0$$. A functional parameter $$\sigma(t)$$, called time symbol, belongs to a certain closed set $$\Sigma\subset C_b(\mathbb{R}, M)$$, where $$C_b(\mathbb{R}, M)$$ denotes the space of bounded continuous functions on $$\mathbb{R}$$ with values in a certain complete metric space $$M$$.
In applications to evolution pde’s a time symbol $$\sigma(t)$$ consists of all time-dependent coefficients, functions and terms from the right-hand side of the evolution equation under consideration. The notion of a process plays the central role: A two-parametric family of mappings $$\{U(t, \tau)\}$$, acting on a Banach space $$E$$, is said to be a process in $$E$$ if the following two conditions are fulfilled: (i) $$U(t, s)\circ U(s, \tau)= U(t, \tau)$$, for all $$t\geq s\geq \tau$$; (ii) $$U(t, t)= \text{Id}$$. The notion of a process generalizes the notion of a semigroup which is usually associated with autonomous evolution equations. Further, the attractor of the family of processes as well as the notion of a complete trajectory and a kernel of a process are defined. The conditions under which a family of processes possesses a compact attractor are investigated.
Using the above general results, the existence of the attractor $$A$$ for the non-autonomous 2D Navier-Stokes system with almost periodic in time external force is proved, as well as the estimate for the Hausdorff dimension of $$A$$.
Reviewer: A.Klíč (Praha)

##### MSC:
 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 58D25 Equations in function spaces; evolution equations 35Q30 Navier-Stokes equations 35K57 Reaction-diffusion equations