cient algorithms for rigorous integration forward in time of dPDEs . Existence of globally attracting xed points of viscous Burgers equation with constant forcing , a computer assisted proof

The dissertation is divided into two separate parts. First part We propose an e cient and generic algorithm for rigorous integration forward in time of systems of equations originating from partial di erential equations written in the Fourier basis. By rigorous integration we mean a procedure, which operates on sets, and return sets which are guaranteed to contain the exact solution. The algorithm is generating, in an e cient way, normalized derivatives, which then are used by the Lohner algorithm to produce a rigorous bound. Algorithm has been successfully tested on several PDEs including the Burgers equation, Kuramoto-Sivashinsky equation and Swift-Hohenberg equation. Problem of rigorous integration in time of partial di erential equations is a problem of large computational complexity, and e cient algorithms are required to deal with PDEs on higher dimensional domains, like the Navier-Stokes equation. Second part We present a computer assisted method for proving the existence of globally attracting xed points of dissipative PDEs. An application to the viscous Burgers equation with periodic boundary conditions and a constant in time forcing function is presented as a case study. We establish the existence of a locally attracting xed point by using computer techniques based on the method of self-consistent bounds. To prove that the xed point is, in fact, globally attracting we introduce a technique relying on construction of an absorbing set, capturing after a nite time any su ciently regular initial condition. Then the absorbing set is rigorously integrated forward in time to verify that any su ciently regular initial condition is in the basin of attraction of the xed point.