Foias, C.; Nicolaenko, B.; Sell, G. R.; Temam, R. Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension. (English) Zbl 0694.35028 J. Math. Pures Appl., IX. Sér. 67, No. 3, 197-226 (1988). 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 Cited in 1 ReviewCited in 102 Documents MSC: 35G05 Linear higher-order PDEs 35K35 Initial-boundary value problems for higher-order parabolic equations 35B10 Periodic solutions to PDEs Keywords:invariant cone property; strong squeezing property; Kuramoto-Sivashinsky equation; inertial Lipschitz manifold; dimension PDFBibTeX XMLCite \textit{C. Foias} et al., J. Math. Pures Appl. (9) 67, No. 3, 197--226 (1988; Zbl 0694.35028)