Evaluating the Dimension of an Inertial Manifold for the Kuramoto-sivashinsky Equation∗

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 ∗This work was supported in part by National Science Foundation Grant Numbers DMS-9706903 and DMS-9705229, CNPq, Braśılia, Brazil, and FAPERJ, Rio de Janeiro, Brazil. Some of the work was completed at the Inst. for Mathematics and its Applications, University of Minnesota. The authors thank Ciprian Foias for a number of helpful suggestions, including Lemma 3.4. Accepted for publication July 1999. AMS Subject Classifications: 35F20, 35Q35, 58F39, 65M70.