  The purpose of this contribution is to show that some of the basic ideas of
turbulence can be addressed in a deterministic setting instead of introducing
random realizations of the fluid. Weak limits of oscillating sequences of
solutions are considered and along the same line the Wigner transform replaces
the Kolmogorov definition of the spectra of turbulence. One of the main issue
is to show that, at least in some cases, this weak limit is the solution of an
equation with an extra diffusion (the name turbulent diffusion appears
naturally). In particular for a weak limit of solutions of the incompressible
Euler equation (which is time reversible) such process would lead to the
appearance of irreversibility. In the absence of proofs, following a program
initiated by P. Lax, the diffusive property of the limit is analyzed, with the
tools of Lax and Levermore or Jin Levermore and Mc Laughlin, on the zero
dispersion limit of the Korteweg-deVries equation and of the Non Linear
Schrodinger equation. The three authors are extremely happy to have the
opportunity to publish this contribution in a volume dedicated to Walter
Strauss as a mark of friendship and admiration for his achievement. They hope
that this paper concerned with non linear fluid mechanics, non linear
instabilities and inverse scattering, will find its place in the different
domains that have interested Walter.
