Zgliczyński, Piotr; Mischaikow, Konstantin Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation. (English) Zbl 0984.65101 Found. Comput. Math. 1, No. 3, 255-288 (2001). A new topological approach for the study of the dynamics of dissipative partial differential equations is presented. The suggested method is based on the concept of self-consistent a priori bounds, which permit the rigorous justification of the use of Galerkin projections. The goal of the investigation is a low-dimensional system of ordinary differential equations subject to small perturbations from the neglected models.For the obtained ordinary differential equations the Conley index is applied to obtain information about the dynamics of the partial differential equations under consideration. The developed method is applied particulary to the Kuramoto-Sivashinsky equation. A computer-assisted proof of the existence of several fixed points in this case is obtained. Reviewer: J.Vaníček (Praha) Cited in 4 ReviewsCited in 45 Documents MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 37B30 Index theory for dynamical systems, Morse-Conley indices 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010) 37L65 Special approximation methods (nonlinear Galerkin, etc.) for infinite-dimensional dissipative dynamical systems 35Q35 PDEs in connection with fluid mechanics Keywords:small perturbation; dynamics of dissipative partial differential equations; Galerkin projections; Conley index; Kuramoto-Sivashinsky equation; computer-assisted proof PDFBibTeX XMLCite \textit{P. Zgliczyński} and \textit{K. Mischaikow}, Found. Comput. Math. 1, No. 3, 255--288 (2001; Zbl 0984.65101) Full Text: arXiv