  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 self-consistent
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. As an application we present a computer assisted proof of the
existence of fixed points for the Kuramoto-Sivashinsky equation.
