  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.
