On the construction of families of approximate inertial manifolds (Q1201116)

The authors propose a method to construct the inertial approximate manifolds (AIM) for nonlinear evolution equations (these are manifolds which attract the orbits in a small (thin) neighbourhood exponentially rapidly). The method is demonstrated with two types of evolution equations: reaction-diffusion equations and the Cahn-Hilliard equation. It is based on decomposition of the solution into a sum the first term of which is the projection of the solution onto the space spanned by the first eigenvectors of the corresponding linear part of the spatial differential operator of the problem; the method is iterative and estimates are constructed successively for expanded projections. Remark. Closeness to AIM is estimated by relations of the type \(| A_ k|\leq M_ k\delta^ k_ k\) where \(\delta_ k\to 0\) as \(k\to\infty\) and \(M_ k\) is bounded for each finite \(k\). The authors note that there are no estimates of the growth of \(M_ k\) as \(k\to\infty\).