Smooth extensions for inertial manifolds of semilinear parabolic equations

TL;DR

This paper investigates the smoothness of inertial manifolds for semilinear parabolic equations, showing that under certain conditions, higher regularity can be achieved by increasing dimension and adjusting nonlinearities, using Whitney extension techniques.

Contribution

It demonstrates that obstacles to higher smoothness of inertial manifolds can be overcome through dimension increase and nonlinearity modification, extending regularity beyond $C^{1, ext{ extvarepsilon}}$.

Findings

Higher regularity of inertial manifolds is achievable under natural assumptions.

Modifying nonlinearity outside the attractor can remove smoothness obstacles.

The Whitney extension theorem is a key tool in the proof.

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 $C^{1,\varepsilon}$-regularity for such manifolds (for some positive, but small $\varepsilon$). Nevertheless, as shown in the paper, under the natural assumptions, the obstacles to the existence of a $C^n$-smooth inertial manifold (where $n\in\mathbb N$ 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 $C^{1,\varepsilon}$-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem.