## Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension.(English)Zbl 0694.35028

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 $$\tilde L=L/(2\pi)$$.
Reviewer: Y.Suyama

### MSC:

 35G05 Linear higher-order PDEs 35K35 Initial-boundary value problems for higher-order parabolic equations 35B10 Periodic solutions to PDEs