Claude Bardos and Basile Nicolaenko

Navier-Stokes Equations and Dynamical Systems

The incompressible Navier Stokes equations are known to be the corner stone of the mathematical theory of fluid mechanic, their study is also an example of the present possibility and limitations of mathematical and numerical tools.

In fact they sit in the middle of hiearchy of equations ranging from Liouville equation for a large system of molecules interacting according a reversible law to very sophisticated model of turbulence used in many engineering sciences ranging from aeronautic design to wheather forcast.

It turns out that this hierachy is by now well understood excluding the level of turbulence modelling. In particular one can explain the appearance of irreversibility. However fully rigourous mathematical proofs are not available in many situations.

This is mainly due to the fact the equations are non linear some basic phenomenas of instabilities can be observed and some mathematical questions like the existence of a global in time smooth solution for the 3d Navier Stokes equation or the 3d Euler equation remain open since the pionnering work of Leray.

Therefore the hierachy of equations and the basic instabilities properties are described in the first part of the article.

In the second part classical examples of equations for turbulent flows are given and it is explained why their derivation is much more difficult than in the first section. One of the main reason being the absence of natural scaling parameter (or separation of scale) as the global Knudsen or Reynolds numbers.

This explains why it may be important to study some asymptotic properties of the Navier Stokes equations related to their large time behaviour and this is the object of the sections 3 and 4.

Even if many other approaches do exist, we will consider only, in the present article, two famillies of objects on one hand the coherent structures for the incompressible 2d Euler equation and on the other hand the notion of global attractors for the forced Navier Stokes equation. The coherent structure correspond to some stable stationnary solutions which are observed in reality (the most classical example being the red spot of Jupiter) . The understanding of their stability is currently under investigation some recents results having been obtained by Miller, Robert, Sommeria and others.

On the other hand if the viscosity is large enough it leads to the concentration of the flow near an object of finite dimension. According to authors one should consider global or exponential attractor. Several authors have studied the ``fractal'' dimension of this attractor and explained how it reduces the Navier stokes equation to an dynamical system in a finite dimensional space.

Eventually some attempt to connect the results of this two last section with turbulence modelling will be described.

Claude Bardos

May 22 1998