Kwak Transform and Inertial Manifolds revisited

TL;DR

This paper revisits the conditions for the existence of inertial manifolds in semilinear parabolic equations, focusing on cases with Jordan cells after applying the Kwak transform, relevant to equations like 2D Navier-Stokes.

Contribution

It provides sharp spectral gap conditions for inertial manifolds in non-self-adjoint cases, especially with Jordan cells after the Kwak transform.

Findings

Sharp spectral gap conditions established

Analysis of Jordan cells in the Kwak transform

Relations between different Kwak transform forms

Abstract

The paper gives sharp spectral gap conditions for existence of inertial manifolds for abstract semilinear parabolic equations with non-self-adjoint leading part. Main attention is paid to the case where this leading part have Jordan cells which appear after applying the so-called Kwak transform to various important equations such as 2D Navier-Stokes equations, reaction-diffusion-advection systems, etc. The different forms of Kwak transforms and relations between them are also discussed.