Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation

Abstract

The Kuramoto-Sivashinsky equation is a dissipative evolution equation in one space dimension which, despite its apparent simplicity, gives rise to a very rich dynamical behavior, as evidenced for instance by the study in [16], of its complicated set of stationary solutions and stationary and Hopf bifurcations. The large time behavior of the solutions is usually embodied by the attractor and the inertial manifolds which have been the object of many studies. In the present article, explicit expressions which can be completely evaluated are obtained for the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation. This involves reworking the analysis in [1] to estimate the radius of the absorbing ball. From there, the choice of phase space, spectral gap condition, and preparation of the equation outside the absorbing ball are varied and the results compared over a moderate domain length. A new preparation of the equation is introduced which leads to the smallest dimension of those compared. The dimension is also obtained for the equation prepared using radii smaller than that of the known absorbing ball, down to the radius of the global attractor suggested by computations.