Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension (Q909097)

In one-space dimension, the Kuramoto-Sivashinsky equation (K-S) can be written as \[ \partial u/\partial t+\partial^ 4u/\partial x^ 4+\partial^ 2u/\partial x^ 2+u(\partial u/\partial t)=0\quad in\quad {\mathbb{R}}\times {\mathbb{R}}_+,\quad u(x,0)=u(x)\quad in\quad {\mathbb{R}}_+, \] u(x\(+L,t)=u(x,t)\) where \(L>0\) and \(u(x,t)=-u(L-x,t)\). L is the size of a typical pattern cell. The author obtains an inertial Lipschitz manifold M for K-S equation and the estimate of the Euclidean dimension, dim M as a function of \(\tilde L;\) dim \(M\leq 1+c_ 1\tilde L^{7/2}\) where \(\tilde L=L/(2\pi)\).