Attractors of periodic processes and estimates of their dimension (Q1907399)

The authors consider abstract nonlinear periodic evolution equations \[ \partial_t u= A(u, t)\quad\text{for}\quad t\geq \tau,\quad u|_{t= \tau}= u_\tau\tag{1} \] on a Banach space \(E\) (\(A(u, t+ p)= A(u, t)\), where \(p> 0\) is the period) and the corresponding to (1) process \(U(t, \tau)\). Using the semigroup \(S(t)\) on the extended phase space \(E\times \mathbb{T}^1\) (\(\mathbb{T}^1\) is the circle \(\mathbb{R}\pmod p\)), which is defined by \[ S(t)(u, \sigma):= (U(t+ \sigma, \sigma)u,\;(t+ \sigma)(\text{mod }p)), \] attractors (i.e. minimal compact attracting sets) for (1) are constructed and their Hausdorff and fractal dimensions are estimated. Further, applications are given to the two-dimensional Navier-Stokes system with periodic external forces as well as to reaction diffusion systems and to dissipative hyperbolic equations with periodic terms.