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

We present a new topological method for the study of the dynamics of dissipative PDE’s. The method is based on the concept of the selfconsistent apriori bounds, which allows to justify rigorously the Galerkin projection. As a result we obtain a low-dimensional system of ODE’s subject to rigorously controlled small perturbation from the neglected modes. To this ODE’s 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 = (u 2)x − uxx − νuxxxx, u(x, t) = u(x+ 2π, t), u(x, t) = −u(−x, t) We obtained a computer assisted proof the existence of the large number fixed points for various values of ν > 0. AMS Subject classification numbers: 37B30, 37L65, 65M60, 35Q35 Research supported in part by Polish KBN grants 2P03A 021 15, 2 P03A 011 18 and NSF–NATO grant DGE–98–04459. Research supported in part by NSF grant DMS-9805584.