Attractors of non-autonomous dynamical systems and their dimension.

*(English)*Zbl 0838.58021This 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 \(A\).

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)