Smooth extensions for inertial manifolds of semilinear parabolic equations

Abstract

The paper is devoted to a comprehensive study of smoothness of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than C 1,ε -regularity for such manifolds (for some positive, but small ε). Nevertheless, as shown in the paper, under the natural assumptions, the obstacles to the existence of a C n -smooth inertial manifold (where n ∈ N is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the C 1,ε -smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem. Contents 1. Introduction 1 2. Preliminaries I: Taylor expansions and Whitney Extension Theorem 6 3. Preliminaries II: Spectral gaps and the construction of an inertial manifold 10 4. Main result 18 5. Verifying the compatibility conditions 25 6. Examples and concluding remarks 30 References 34

A Unified Theory for Inertial Manifolds, Saddle Point Property and Exponential Dichotomy

“…Continuity of the invariant (inertial if δ > 0) manifold with respect to perturbations on f , and thus continuity with respect to parameters, follows as usual; see [8] for a detailed proof in the case that ρ < 0. In contrast with the usual regularity issues for stable/unstable manifolds, when the C k regularity of the vector field implies the C k regularity of these invariant manifolds, the regularity of invariant (inertial if δ > 0) manifolds is more delicate, see [13,23]. However, such regularity can be achieved by stronger exponential gap conditions.…”