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

该提案的研究部分涉及基本的数学问题，涉及不可压缩流体的非线性偏微分方程偏微分方程，色散（保守，波动）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、纳维尔-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弗拉索夫-麦克斯韦方程组。它们都出现在科学和工程中，作为动力学过程的基本模型。应用领域包括量子物理学、广义相对论、光学、等离子体物理学、流体湍流、微物理学、天气预报和边界层。 虽然这个项目的动机是许多经典的问题，已经研究了世纪，这个建议的总体目标是获得分析信息，我们确定了一些新的问题。 这些问题本质上是有趣的，从数学和物理的角度来看，有关问题的适定性，不稳定性，长时间的行为，和奇点的形成。此外，我们提出的问题也将揭示在该领域的一些主要的经典问题，有关的整体适定性与爆破的强解和适定性与非唯一性和不稳定性的弱解。

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.