Low-dimensional chaos and asymptotic time behavior in the mechanics of fluids

We discuss here Chaotic Advection in laminar incompressible flows and long-time diffusive behavior. The basic mechanism for the chaotic behavior generated by homoclinic intersections in quasi-integrable Hamiltonian systems, as well as the multi-scale expansion, turn out to be still the main ingredients of this issue. The multi-scale approach for partial differential equations, which will be explained below, can be seen as a spatio-temporal extension of the methods introduced by Lindstedt, and improved by Poincar´e, for the treatment of the secular terms in Mechanics.