Attractors and Finite-Dimensional Behaviour in the 2D Navier-Stokes Equations

The purpose of this review is to give a broad outline of the dynamical systems approach to the two-dimensional Navier-Stokes equations. This example has led to much of the theory of infinite-dimensional dynamical systems, which is now well developed. A second aim of this review is to highlight a selection of interesting open problems, both in the analysis of the two-dimensional Navier-Stokes equations and in the wider field of infinite-dimensional dynamical systems. 1. Introduction The Navier-Stokes equations are the fundamental mathematical model of fluid flow; a physical derivation of the equations can be found in Batchelor [1] or Doering and Gibbon [2], for example. Their rigorous analysis goes back to Leray [3], who proved the global existence of weak ( -valued) solutions in 3D and local existence of strong ( -valued) solutions; similar results were obtained by Hopf [4] for bonded domains. Global existence and uniqueness of weak solutions in the 2D case was first shown by Ladyzhenskaya [5]. The dynamical systems approach to the Navier-Stokes equations was developed over a number of years, notably by Ladyzhenskaya [6] and Foias, Constantin, Temam, and coauthors; see Constantin et al. [7], for example. Delay differential equations provided a stimulus for the development of the theory from a different but related viewpoint; see Hale et al. [8], for example. Since rigorous existence and uniqueness results are only available for the 2D equations, we confine ourselves here to this case. The Navier-Stokes equations are posed on a spatial domain , supplemented with appropriate boundary conditions. Here is the two-component velocity, the parameter is the kinematic viscosity, and is the scalar pressure, which serves to enforce the divergence-free condition . The right-hand side is a (somewhat artificial) “body force,” which serves to maintain some nontrivial motion of the fluid. For simplicity we will treat the equations on a periodic domain , so that , where are unit vectors parallel to the coordinate axes. In addition we will make the simplifying assumption that and have zero average over , and that is divergence-free ( ). Although we will generally confine our analysis to the case of periodic boundary conditions, many results are also true for the case of Dirichlet boundary conditions, and we will occasionally comment on this case in what follows. Note that while much of the existence and uniqueness theory, particularly in the 3D case, is carried out in the whole space setting (which allows one to use the tools of harmonic analysis; see e.g.,