Self-consistent bounds method for dissipative PDEs

We discuss the method of self-consistent bounds for dissipative PDEs with periodic boundary conditions. We prove convergence theorems for a class of dissipative PDEs, which constitute a theoretical basis of a general framework for construction of an algorithm that computes bounds for the solutions of the underlying PDE and its dependence on initial conditions. We also show, that the classical examples of parabolic PDEs including Kuramoto-Sivashinsky equation and the Navier-Stokes on the torus fit into this framework.