Rigorous Numerics for Partial Differential Equations: The Kuramoto-Sivashinsky Equation

Piotr Zgliczyński, Konstantin Mischaikow

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76 Scopus citations


We present a new topological method for the study of the dynamics of dissipative PDEs. The method is based on the concept of the self-consistent a priori bounds, which permit the rigorous justification of the use of Galerkin projections. As a result, we obtain a low-dimensional system of ODEs subject to rigorously controlled small perturbation from the neglected modes. To these ODEs we apply the Conley index to obtain information about the dynamics of the PDE under consideration. We applied the method to the Kuramoto-Sivashinsky equation ut = (u2)x - uxx - vuxxxx, u(x, t) = u(x + 2π, t), u(x, t) = -u(-x, t). We obtained a computer-assisted proof of the existence of several fixed points for various values of v > 0.

Original languageEnglish (US)
Pages (from-to)255-288
Number of pages34
JournalFoundations of Computational Mathematics
Issue number3
StatePublished - Aug 2001
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


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