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

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 = (u²)_x - u_xx - vu_xxxx, 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.