Analysis of Mathematical models for the ocean, atmospherics sciences and optics.

这个项目涉及从量子力学、几何光学和流体力学中产生的偏微分方程式的分析。更具体地说，一方面介绍了非线性薛定谔方程和非线性波动方程(作为色散方程的模型)，另一方面介绍了Navier-Stokes、Boussinesq、原始方程和相关的流体模型。 尽管流体模型的建模、物理和计算部分都得到了很好的发展，但数学研究似乎并没有跟随同样的进展。与此同时，在过去的二十年里，色散偏微分方程解的分析也取得了巨大的进展。因此，人们如何使用和利用这些新的技术和工具来进一步分析来自流体力学的偏微分方程组，这是一个自然的问题。在最近的几次尝试中，人们已经可以观察到这一点的开始。因此，这个项目的总体目的是汇集这些看似不同类型的偏微分方程分析的想法和技术，以便在对湍流和许多其他相关物理现象的数学理解方面取得进展。

This project concerns the analysis of Partial Differential Equations arising from quantum mechanics, geometric optics and fluid mechanic. More specifically, it focuses on the one hand on, the Nonlinear Schrödinger and Nonlinear Wave equations (as models for dispersive equations), and on the other hand on, the Navier-Stokes, Boussinesq, the primitive equations and related fluid models. Although the modeling, physical and the computational parts were very well developed for fluid models, the mathematical study did not seem to follow the same stream of progress. Meanwhile and during the last two decades, a tremendous progress has been made in the analysis of dispersive partial differential equations. So that comes the natural question on how one can use and take advantage of those new techniques and tools to further analyze the partial differential equations coming from fluid dynamic. One can already observe the beginning of this in several recent attempts. So the general purpose of this project is to bring together ideas and techniques from these seemingly different kinds of Analysis of PDEs in order to make progress on the mathematical understanding of turbulence and many other relevant physical phenomena.