This dissertation is intended to contribute to the analytical understanding of nonlinear dissipative partial differential equations. It is centred around the investigation of the Complex Ginzburg-Landau equation with a nonlinear term of arbitrary order. The work contains the following main section.
Introduction of the analytical framework: Presentation of the main concepts for an abstract evolution equation. Starting from a description of the functional setting, attractors, attractor regularity, bounds on the attractor dimension and finally inertial manifolds are introduced.
Discussion of the Complex Ginzburg-Landau equation: The abstract concepts are applied to the Complex Ginzburg-Landau equation with periodic boundary conditions on a spatial domain of dimension d. All proofs are explicitly done with a nonlinear term of order 2q+1. One of the results is that the quantity qd has crucial influence on the structure of all but the inertial manifold results. Furthermore two regions in the paramenter plane in which the estimates of the sup-norm are qualitatively different are established and a physical interpretation in terms of soft (weak) and hard (strong) turbulence is suggested. The present work generalises previous results by Bartuccelli, Constantin, Doering, Gibbon and Gisselfält (1990). The so-called ladder theorem of these earlier studies is abandoned in favour of a more flexible differential inequality, thereby shortening the proofs and sharpening the estimates in some cases.
The physical notion of length scales is introduced and linked to the previous concepts via the discussion of relevant modes. The Complex Ginzburg-Landau Equation is used as a test case in the examination of these ideas.
A German translation of this dissertation is also available.