Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations

We evaluate several alternative methods for the approximation of inertial manifolds for the one-dimensional Kuramoto-Sivashinsky equation (KSE). A method motivated by the dynamics originally developed for the Navier-Stokes equation is adapted for the KSE. Rigorous error estimates are obtained and compared to those of other methods introduced in the literature. Formal relationships between these other methods and the one introduced here are established. Numerical bifurcation diagrams of the various approximate inertial forms for the KSE are presented. We discuss the correspondence between the rigorous error estimates and the accuracy of the computational results. These methods can be adapted to other dissipative partial differential equations.