Inertial manifolds via spatial averaging revisited

TL;DR

This paper provides a comprehensive analysis of inertial manifolds for various semilinear parabolic equations in 3D, using the spatial averaging method, and introduces new results and applications in fluid dynamics and phase transition models.

Contribution

It offers a universal approach to inertial manifolds via spatial averaging, covering existing results and presenting new findings for complex equations in 3D.

Findings

Unified framework for inertial manifolds in 3D equations

Extension to fractional and higher-order Cahn-Hilliard equations

Applications to modified Navier-Stokes equations with regularizations

Abstract

The paper gives a comprehensive study of inertial manifolds for semilinear parabolic equations and their smoothness using the spatial averaging method suggested by G. Sell and J. Mallet-Paret. We present a universal approach which covers the most part of known results obtained via this method as well as gives a number of new ones. Among our applications are reaction-diffusion equations, various types of generalized Cahn-Hilliard equations, including fractional and 6th order Cahn-Hilliard equations and several classes of modified Navier-Stokes equations including the Leray-$\alpha$ regularization, hyperviscous regularization and their combinations. All of the results are obtained in 3D case with periodic boundary conditions.