Equations aux dérivées partielles dispersives – Equa-disp

The central topic in our project is the study of the properties of solutions to nonlinear dispersive equations. This category represents a large number of PDE of hyperbolic type. Most known examples are wave, Schrodinger, Korteweg-De-Vries, Benjamin-Ono equations. All these equations have the particularity of dispersing the waves. In the 80's, with the works of Ginibre-Velo, Bourgain, Kapitanski, Kato (and others), the discovery of this kind of properties, which can be seen (modulo a time averaging) as a regularity gain, proved to be crucial in solving these non linear equations. Whereas the studies around these type of equations have been considerably developed, many progress are still to be done: Our project can be divided into 4 topics. It is however clear that this division is arbitrary since these topics are related to each other and that any progress on one of the following topics could result in a progress in another. A: The longwave dispersive nonlinear PDE's (such as KdV, Benjamin-Ono, KP,... equations) are 'universal' asymptotic models to describe the long time dynamics of various complicated wave propagation phenomena in the weakly nonlinear long wave regime. For instance they are models for (surface or internal) water waves, for plasma waves, for waves in ferromagnetic materials, etc... They occur also as the long wave (transonic in the case of the Gross-Pitaevskii equation) limit of the nonlinear Schrödinger equations. One important issue is the rigorous justification of the models. Another fundamental issue is the comparison (by BKW techniques) between the solutions of the original system (in the right scaling) with those of the models, with the correct error estimates. An important issue, both for modelling and theoretical considerations, is that of the control theory for dispersive long wave models which leads o numerous open questions despite the recent work of J. M. Coron, L. Rosier and their collaborators. All these issues cannot ignore the numerical simulations which often are the only guide to get some insights on the dynamics. B: The influence of the geometry of the medium where the propagation occurs is still very poorly understood, in particular for Schrödinger type equations for which the speed of propagation is infinite, and consequently the global geometry is seen instantaneously. The geometry for which a satisfactory answer is known are very few. These are essentially the tora (Bourgain's work) and the spheres. Another kind of geometric influence can appear with boundary value problems. In these case, even for (constant coefficient) wave equations, the known results are very partial C: The relationship between Partial differential equations and probability theory is clearly a very important field. On one hand, it allows to model some physical phenomena via stochastic noise forces and on the other hand it might allow on certain topics to go beyond the deterministic approach: Indeed in many cases, the obstruction to solving some dispersive equations or the obstructions to proving stability results are well known and appear to be 'exceptional events' which might be dealt with via a probabilistic approach D: In general relativity, (quasi linear wave equations), some particular solutions to the Einstein equations are well known (Minkovski, Schwarzschild and Kerr metrics). However, except for the case of Minkovski metric (work by Klainerman and Christodoulou), the stability of these solutions is still an open problem. Another kind of question to be addressed concerns the case of small (quasi linear or semi-linear) perturbations of wave equations in the whole space. In this case,the Cauchy theory is now well understood. However, the large time behavior : length of the time interval of existence, influence of the non linearity (short, large, or even very large range), scattering properties, wave operators, etc, is still largely to be understood. The goal of this project is to gather skills around these themes