Approximate inertial manifolds of exponential order (Q1355013)

An approximate inertial manifold (AIM) is a smooth finite dimensional manifold of the phase space which attracts all orbits to a thin neighborhood of it in a finite time uniformly for initial conditions on a given bounded set. This neighborhood contains the global attractor. An AIM is said to have order \(\eta\) when its associated neighborhood has width \(\eta\). Let \(\lambda_n\) be the \(n\)th distinct eigenvalue of the Stokes operator. \textit{C. Foias} and \textit{R. Temam} [SIAM J. Math. Anal. 25, No. 5, 1269-1302 (1994; Zbl 0836.35118)] constructed a family \(\{{\mathcal M}_j\}^\infty_{j=0}\) of AIMs with order \(\eta\sim K(2^{-j} +e^{-\delta \lambda_{n+1}})\), \(\delta >0\). The author presents a construction of another family of AIM s which has also exponential order with respect to \(j\) and \(\lambda_n^{1- \alpha}\), \(0\leq \alpha \leq 1\) and converges even exponentially to the exact inertial manifold when the spectral gap condition is satisfied. The construction is based on a variation of the Lyapunov-Perron method. The presented construction can be applied for examples to the Navier-Stokes, Kuramoto-Sivashinsky and Cahn-Hilliard equations.