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

与流体动力学、地球物理流、量子物理和光学相关的非线性偏微分方程中的渐近和奇点形成。

基本信息

批准号：
RGPIN-2019-06422
负责人：
Ibrahim, Slim
金额：
$ 1.38万
依托单位：
University of Victoria
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2020
资助国家：
加拿大
起止时间：
2020-01-01 至 2021-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715677
关键词：
Asymptotics singularity formation Nonlinear PDEs

项目摘要

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.

该提案的研究组成部分解决了不可压缩流体，分散性（保守，波浪状）和动力学模型的非线性部分微分方程PDE的基本数学问题。示例包括非线性Schroedinger和非线性波方程，Euler，Navier-Stokes，原始方程（也称为Hyderstatic Euler），Prandtl方程以及相对论Vlasov-Maxwell系统。它们在整个科学和工程中都出现，作为动力学过程的基本模型。应用领域包括量子物理，一般相对论，光学，血浆物理学，流体湍流，地球物理学，天气预测和边界层。尽管该项目的动机是许多经典问题已经研究了一个多世纪，但该提案的总体目的是获取有关我们确定的许多新问题的分析信息。从数学和物理观点角度来看，这些问题在本质上都是有趣的，这些观点与适应性良好，不稳定性，长期行为和奇异性形成有关。此外，我们提出的问题还将阐明该领域的某些主要经典问题与全球良好性和爆炸有关，以实现强大解决方案，适应性良好，无唯一性和弱解决方案的不稳定。