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Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation. (English) Zbl 0984.65101

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.

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
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