In this survey, we shall mainly consider dynamical systems or semiflows generated by partial differential equations of autonomous type. We shall also mostly restrict our attention to scalar equations defined on bounded domains.
In a first part, we shall present classical existence results of compact global attractors A for asymptotically compact, dissipative dynamical systems. We shall also discuss basic properties such as connectedness and stability. The dynamics on the global attractor A can be very complicated, and in general cannot be described. This complexity is partly reflected in the dimension of A.
We shall review a few properties which allow a better knowledge of the dynamics on the global attractor A. If the dynamical system is gradient that is, has a strict Lyapunov function, A is the unstable set of the equilibrium points; in particular, a Morse decomposition of A can be given. We shall describe more precisely the dynamics on A, for a class of scalar reaction-diffusion equations (such as the Chafee-Infante equation). Some global attractors are contained in C1-submanifolds of finite dimensions, called inertial manifolds, which should allow to reduce their study to a finite dimensional system of differential equations.
Partial differential equations often depend on parameters like the domain of definition of the equation, diffusion coefficients, different types of boundary coefficients, etc... Thus a question of interest is the study of the dependence of the attractor and the flow on it in these parameters. Some continuity properties of the attractors with respect to parameters and structural stability results will be described.
All the above topics will be illustrated with examples, including reaction-diffusion equations, wave equations with damping, Navier-Stokes, Cahn-Hilliard, Kuramoto-Sivashinsky equations, etc...
Université de Paris-Sud
May 18 1998