Rigorous study of dynamics of dissipative PDE's

The goal of this project is to develope the tools for rigorous study of the dynamics of dissipative PDE's (Kuramoto-Sivashinsky (KS), Navier-Stokes (NS), Ginzburg-Landau (GL)), etc. The method combines topological methods and rigorous numerics.

General description of the project with applications to KS equations

A conference presentation in form of slides ps

A paper with K. Mischaikow with the description of the method: ps . This paper contains a proof of the existence of multiple steady states for KS-eq. The issue of local uniqueness of these fixed points and a general question how to produce a rigorous steady state bifurcation diagram is treated in paper Towards rigorous bifurcation diagram .. . It contains essentially all theory and formulas necessary for this purpose and the proof of the existence of several bifurcations - the numerical data from these proofs are given here

In paper Attracting fixed points for Kuramoto-Sivashinsky equation using the estimates for Lipschitz constants based on logarithmic norms introduced in the paper 'Trapping regions ...' (see below in NS-section) we show that some of the fixed points whose existence was established in paper with K. Mischaikow are attracting. The text file appendix contains numerical data from proofs.

Recently, I have proved the existence of periodic orbits, both stable and unstable ones, for KS-equations for parameter values of mu in the range [0.02991,0.128]. The proof combines self-consistent bounds with rigorous numerics for PDEs. pdf the numerical data from these proofs are given here , see also earlier paper on this topic ps

Navier-Stokes Equations

The paper Trapping regions and ODE-type proof of the existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane gives a self-contained account of the ODE-type proof by Mattingly and Sinai (see also a paper by E and Sinai ) of the existence and uniqueness of the Navier-Stokes systems with periodic boundary conditions on the plane. Mattingly and Sinai called their proof elementary, but the their proof was ODE-type (elementary in their sense) only up to the moment of getting the trapping regions for all Galerkin projections, but to pass to the limit with the dimension of Galerkin projections they invoked classical results from PDE theory (which for sure are not elementary in any sense), which are usually not mastered by the researchers working in dynamics of ODE's, to which this note is mainly addressed. In this note we fill-in this gap by giving ODE-type arguments, which allow to pass to the limit. Using ODE-type estimates based on the logarithmic norms we also obtained uniqueness and an estimate for the Lipschitz constant. In fact we have proved (using ODE-tools only) that on the trapping region we have a semidynamical system. The results we prove here are well known for Navier-Stokes system in 2D, but the method of getting very nice estimates on Galerkin projections presented here appears to be new.

Convergence of Galerkin projections

The paper Trapping regions and ODE-type proof of the existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane gives nice argument for convergence of Galerkin projections for NS-2D, KS-1D or NS-3D (with small initial data and force). A paper On smooth dependence on initial conditions for dissipative PDEs goes beyond this and gives a proof that also the derivatives with respect to initial conditions for Galerkin projections converge. This gives rise to smooth dependence on initial conditions for full PDEs on the trapping regions. The proof does not give rate of convergence, so we cannot use it in the rigorous numerics for PDEs get some hyperbolicity, but I hope that soon I will do this too.

Rigorous numerics

The description of C^1 and C^0 Lohner algorithm explaining how we do rigorous computations for ODE's is here: pdf . This paper contains also proofs of the existence of attracting periodic orbit for Rossler equations for a=2.2 and other close values, and for 14-dim Galerkin projection of the Kuramoto-Sivashinsky equations.

The description of an algorithm, which I use to compute the Poincare maps for PDE's can be found here ) (see also ps ). This is a modification of Lohner algorithm to ODE's with perturbation. An ODE is the Galerkin projection of PDE and the perturbation is given by self-consistent a-priori bounds, but a the tail evolves this perturbation changes in time.

Infinite dimensional discrete systems

The paper with Z. Galias A rigorous numerical ... contains a generalization of Krawczyk method to the context of infinite dimensional disipative system. In this paper it is applied to an intergral map - Kot-Schaffer dispersal-growth model. We treat this a preparation of C^1-tools for rigorous investigation of dissipative PDEs.

Last modified December 9, 2004