Weak Convergence and Deterministic Approach to Turbulent Diffusion

Abstract

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.