Asymptotics and singularity formation in Nonlinear PDEs related to fluid dynamic, geophysical flows, quantum physics and optics.

该提案的研究部分解决了基本的数学问题，涉及不可压缩流体的非线性偏微分方程、色散（保守的、波状的）和动力学模型。例子包括非线性薛定谔方程和非线性波动方程、欧拉方程、纳维-斯托克斯方程、原始方程（也称为流体静力学欧拉方程）、普朗特方程和相对论性的弗拉索夫-麦克斯韦方程组。它们都作为动态过程的基本模型出现在整个科学和工程中。应用领域包括量子物理、广义相对论、光学、等离子体物理、流体湍流、地球物理学、天气预报和边界层。虽然这个项目的动机是许多已经研究了一个多世纪的经典问题，但这个建议的总体目标是获得关于我们发现的一些新问题的分析信息。从数学和物理的角度来看，这些问题本质上是有趣的，这些问题与适定性、不稳定性、长期行为和奇点形成有关。此外，我们提出的问题还将揭示该领域中与强解的全局适定性与爆破以及弱解的适定性与非唯一性和不稳定性相关的一些主要经典问题。

The research component of the proposal addresses fundamental mathematical problems invoking nonlinear partial differential equations PDEs of incompressible fluids, dispersive (conservative, wave-like) and kinetic models. Examples include the nonlinear Schroedinger and nonlinear wave equations, Euler, Navier-Stokes, primitive equations (also known as hydrostatic Euler), Prandtl equation, and the relativistic Vlasov-Maxwell system. They all arise throughout science and engineering as fundamental models of dynamical processes. Areas of application include quantum physics, general relativity, optics, plasma physics, fluid turbulence, geophysics, weather prediction and boundary layers. While this project is motivated by many classical questions which have been studied for over a century, the overall objective of this proposal is to obtain analytical information about a number of new problems that we identify. These problems are intrinsically interesting from both the mathematical and physical points of view relating issues of well-posedness, instabilities, long-time behaviour, and singularity formation. Moreover, the problems we are proposing will also shed light on some of the major classical problems in the field related to global well-posedness vs. blow-up for strong solutions and well-posedness vs. non-uniqueness and instability for weak solutions.