ANAÉ

The last decades has witnessed a very fast and deep development in the field of evolution partial differential equations (and particularly the dispersive equations). These major advances allow some perspectives which appeared to be completely out of reach a few years ago and open the very exciting perspective of studying deep dynamical properties of solutions of Partial differential equations. On the other hand, specialists of dynamical systems successfully extended methods and ideas, developed for the study of finite dimensional models, to infinite dimensional ones. This led to spectacular results concerning long time behavior of solutions of some non-linear partial differential equations, especially in one space dimension. The tools developed in the PDE context to handle non-linear PDEs could also lead to major breakthroughs, when combined with some dynamical properties of the equations. Finally, the study of partial differential equations in the presence of randomness, a topic originating in ideas from statistical mechanics, has also recently seen spectacular results. The goal of this project is to blend together ideas from these three points of view, to gain some new insights on the behaviour of solutions of partial differential equations in some asymptotic regimes : -- Long time behaviour, -- Existence, scattering, -- Stabilities, instabilities of particular solutions -- Blow-up.