Estimates on the lowest dimension of inertial manifolds for the Kuramoto-Sivasbinsky equation in the general case

Abstract

We derive estimates on the lowest dimension in various Sobolev spaces of inertial manifolds for the Kuramoto-Sivashinsky equation: ∂u/∂t+ vD4u+D2u+uDu=0 for solutions which are periodic with period L. Contrary to earlier results in [3] and other works, there is no requirement on the antisymmetry of the initial data. Our results are: 1. the lowest dimension of inertial manifolds in the Sobolev space Hm is bounded by a universal constant times L0.82m+2.05; 2. the lowest dimension of inertial manifolds in L2 is bounded by a universal constant times L1·64(In L)0.2 where. © 1994, Khayyam Publishing.