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Foundations of Computational Mathematics

, Volume 1, Issue 3, pp 255–288 | Cite as

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

  • Piotr Zgliczynski
  • Konstantin Mischaikow
Article

Abstract

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
$$u_t = \left( {u^2 } \right)_x - u_{xx} - vu_{xxxx} , u(x,t) = u(x + 2\pi , t), u(x,t) = - u( - x,t).$$

We obtained a computer-assisted proof of the existence of several fixed points for various values of ν > 0 .

AMS Classification. 37B30, 37L65, 65M60, 35Q35. 

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Copyright information

© Society for the Foundation of Computational Mathematics 2000

Authors and Affiliations

  • Piotr Zgliczynski
    • 1
  • Konstantin Mischaikow
    • 2
  1. 1.Jagiellonian University Institute of Mathematics Reymonta 4 30-059 Kraków, Poland zgliczyn@im.uj.edu.plPL
  2. 2.Center for Dynamical Systems and Nonlinear Studies School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, USA mischaik@math.gatech.eduUS

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