The generalized complex Ginzburg-Landau (CGL) equation describes the evolution of a complex-valued field. It has a long history in physics as a generic amplitude equation near the onset of instabilities that lead to chaotic dynamics in fluid mechanical systems, as well as in the theory of phase transitions and superconductivity. In this role it and equations like it have had remarkable success in describing evolution phenomena in a broad range of physical systems, from fluids to optics. More recently, it has been proposed and studied as a model for ``turbulent'' dynamics in nonlinear partial differential equations. It is a particularly interesting model in this respect because it is a dissipative version of the nonlinear Schrödinger equation, a Hamiltonian equation which can possess solutions that form localized singularities in finite time.
This article summarizes the status of well-posedness and regularity questions for the CGL equation subject to periodic boundary conditions in any spatial dimension. It also discusses the relationship of these results to the above investigations and, in particular, to analogous issues for Navier-Stokes fluid turbulence. Appropriately defined weak solutions exist globally in time and unique classical solutions are found locally. Conditions are given under which these classical solutions exist globally in time and have bounds on all derivatives that are uniform over large times, thereby providing the setting in which ideas from finite dimensional dynamical systems can be applied to the CGL equation. Moreover, by using a characterization of Gevrey classes in terms of decay of Fourier coefficients, these solutions are shown to be analytic for positive times. Refinements in both some of the theorems and some of the proofs are introduced.