Asymptotic dynamics for nonlinear dispersive systems

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At present, some simple "elementary" systems start to be reasonably well understood. However, passing from the understanding of microscopic objects to moderately complex configurations remains a challenging problem. This projects aims at understanding simple but basic questions for slightly complicated systems, focusing on situations and models related to macroscopic plasma physics. A plasma is a collection of many charged particles that, because of their long range interactions exhibit a collective macroscopic behavior which is complex and remains poorly understood. This in turns limits some potential major applications, controlled fusion being perhaps the most well-known. Thus, furthering the understanding of this "small scale to large scale" cascade of information can have major possible applications. The emphasis of this proposal is on large-time questions such as "can a plasma macroscopically at rest spontaneously develop dramatic behavior such as high concentration or velocities, or spontaneously form a vacuum?". Another related question is to find possible scenarios for the large-time behavior of a plasma that remains away from the quiet neutral equilibrium. Investigating these questions involve a fine study of the deep mechanisms of transfer of energy inside different parts of the plasma as a result of collective elementary simple interactions.This projects aims at developing the study of two fundamental equations from physics. First we consider stability issues for the 2-fluid Euler-Maxwell equation in two dimensions. This is a fundamental equation modeling the dynamical properties of a plasma. The goal is to prove that under certain conditions, small perturbations of an equilibrium will not develop shocks and that actually, the plasma will get back to equilibrium, even in the absence of dissipation. Such a result would be of great physical and mathematical importance as it is known to be false for the compressible Euler equation in the absence of a self-consistent electromagnetic field. In a second part, we study the Schroedinger equation on a curved background. This equation is a universal model appearing in many time reversible equations, notably in some plasma models. When this equation is posed on a nonconstant background, many classical tools from the Euclidian theory break down and one expects the appearance of various new phenomena due to the influence of the geometry. We study in particular the effect of the growth of the volume on the global behavior of the solutions and on the possibility to obtain asymptotic dynamics different from scattering. A unifying theme is to understand the asymptotic dynamics of solutions in a context where the dispersion is limited. Another unifying theme is to try to find simpler limit equations and understand their significance for the full model. A last unifying theme concerns the specificity and relevance of dispersive systems. Some tools we plan to use and develop are concentration compactness methods, study of the space and time resonances, study of dispersive systems and pseudo-products estimates with singular multipliers.