Lectures given at

Departamento de Ecuaciones Diferenciales y Analisis Numerico

Universidad de Sevilla

Easter 1997

These notes discuss various aspects of the long-time behaviour of the solutions of the incompressible Navier-Stokes equations. Some of these properties are analytical in nature, and others rely on the consideration of the solutions generated by the equation as a dynmaical system in a suitable infinite-dimensional phase space.

Existence and uniqueness results are covered in chapter one; essentially it is shown how to use define a dynamical system from the solutions of the equations for the two-dimensional case. With some regularity assumptions, one can do the same for the three-dimensional case.

Chapter two considers the existence of global attrators, in both the 2D and 3D situations -- in the latter case, both with regularity assumptions, and some recent developments due to Sell which cope with the possible lack of uniqueness in this case.

Chapter three introduces the squeezing property, which can be used to show that the attractors are finite-dimensional, and also to show that a finite number of Fourier modes are ``determining'' (in a way which is made precise). Chapter four shows, similarly, that a sufficiently closely spaced number of spatial observations, or volume averages, are also ``determining''.

Chapter five briefly discusses exponential attractors, inertial manifolds, and approximate inertial manifolds. Only exponential attractors and approximate inertial manifolds are known to be applicable to the Navier-Stokes equations.

Chapter six discusses ways of embedding the dynamics on the finite-dimensional attractor from chapters two & three into a dynamical system on a finite-dimensional space, and shows that the dynamical can be approximated by a system on a three-dimensional space.

Finally, a full list of references is available.