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：
2022
资助国家：
加拿大
起止时间：
2022-01-01 至 2023-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759002
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.

该提案的研究部分解决了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、波状）和动力学模型的基本数学问题。示例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、本原方程（也称为静水欧拉方程）、普朗特方程和相对论弗拉索夫-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领域。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、地球物理学、天气预报和边界层。虽然该项目的动机是许多已经研究了一个多世纪的经典问题，但该提案的总体目标是获得有关我们发现的许多新问题的分析信息。从数学和物理角度来看，这些问题本质上都很有趣，涉及适定性、不稳定性、长期行为和奇点形成等问题。此外，我们提出的问题还将阐明该领域的一些主要经典问题，这些问题涉及强解的全局适定性与爆炸性以及弱解的适定性与非唯一性和不稳定性。