Inertial Manifolds via Spatial Averaging Revisited
Anna Kostianko, Xinhua Li, Chunyou Sun, Sergey Vital'evich Zelik · SIAM Journal on Mathematical Analysis · 2022
The paper gives a comprehensive study of inertial manifolds for semilinear parabolic equations and their smoothness using the spatial averaging method suggested by Sell and 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 sixth 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 three-dimensional case with periodic boundary conditions.