Exponential attractors for nonautonomous first-order evolution equations
The authors consider an abstract first-order evolution problem of the form \[ {du\over dt}+ Au+ R(t,u)= 0,\tag{1} \] \[ u(\tau)= u_\tau,\quad \tau\in \mathbb{R},\tag{2} \] with \(u\in E\), \(E\) being a Banach space. They rewrite (1) in the form \[ {du\over dt}= G_{\delta_0(t)}(u),\tag{3} \] where \(\delta_0(t)\) (called the time symbol of the equation) is quasiperiodic with respect to the time. It means that \(\delta_0(t)= \delta_0(\alpha_1 t,\dots,\alpha_k t)\), where \(\alpha_1,\dots,\alpha_k\) are rationally independent and \(\omega_i\to \delta_0(\omega_1,\dots, \omega_k)\) is \(2\pi\)-periodic, \(i= 1,\dots, k\). The authors associate to the problem (3), (2) a family of problems (4), (2), with \[ {du\over dt}= G_{\delta(t)}(u)\tag{4} \] and \(\delta\in\Sigma\), \(\Sigma= \{\delta_0(\alpha_1 t+\omega_{01},\dots, \alpha_k t+\omega_{0k})\), \((\omega_{01},\dots, \omega_{0k})\in T^k\}\), \(T^k\) denotes the \(k\)-dimensional torus. Under suitable assumptions the authors construct a semigroup \(S(t)\) on the extended space \(E\times\Sigma\) and prove that \(S(t)\) possesses a compact exponential attractor. Next, they study its projection on \(E\) and establish that it is a compact, exponentially attracting set with finite fractal dimension called it a uniform exponential attractor. Then they apply the above abstract approach to the two-dimenisonal Navier-Stokes equation on \((0,L)^2\) with periodic boundary conditions.
scientific article; zbMATH DE number 1492526