Regularity, Stability, and Uniqueness Questions for Certain Non-Linear Partial Differential Equations

某些非线性偏微分方程的正则性、稳定性和唯一性问题

基本信息

批准号：
1956092
负责人：
Vladimir Sverak
金额：
$ 34.95万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1956092&HistoricalAwards=false
关键词：
Regularity Stability Uniqueness Questions Certain

项目摘要

Partial Differential Equations (PDEs) are at the basis of many mathematical models used in science and engineering. Examples include the equations for fluid flows or the equation describing the distribution of stress in various structures. In practice, the equations are often solved with the use of computers and a good theoretical understanding of the equations is important for finding effective algorithms. At present, our theoretical understanding of many PDEs is incomplete. There is an important difference between linear models (for which our understanding is better) and non-linear models. In linear models, the reaction of the system to a disturbance is, roughly speaking, proportional to the disturbance. In non-linear models, this is not the case, and many of the mathematical difficulties can be traced to this effect. Linear regimes are often relevant for small disturbances from equilibria, whereas large disturbances are often governed by non-linear phenomena. This project will focus on the non-linear phenomena. In particular, one of the most serious effects in the class of equations which will be investigated is the formation of singularities and the related loss of predictive power of the equations. This will be studied in the context of fundamental equations (such as the equations of incompressible fluid mechanics) and also for various model equations, which can provide suitable stepping stones towards making progress on difficult open problems. This project provides research training opportunities for graduate students.At a more technical level, the project focuses on the following areas: (i) One-dimensional models exhibiting features of PDEs of fluid mechanics. These include the De Gregorio model (which can be thought of as an extension of the Constantin-Lax-Majda model), equations modeling boundary behavior of 2d systems, and vector-valued Burgers-type equations. In spite of their simplicity, such models can be a good source of ideas and their improved understanding can lead to progress on the fundamental equations. In fact, ideas going back to these models have already proved important in the context of the full three-dimensional incompressible Euler equations; (ii) Equations arising in physics and geometry for which we have a satisfactory chance of obtaining a fairly good understanding. These include the Complex Ginzburg-Landau equation, the 2d harmonic map heat flow, and some classical non-linear elliptic systems arising from multi-dimensional variational integrals for vector-valued functions. (The last theme has connections to non-linear elasticity.) All these are important equations in their own right and the PI believes that some of the long-standing open problems related to them can be successfully addressed; (iii) Approachable aspects of the full equations of the incompressible fluid dynamics (the Navier-Stokes and Euler equations). This includes investigations of the solvability of equations describing generalized self-similar singularities, relations between possible non-uniqueness of the Leray-Hopf solutions and questions about instabilities, the stability of the double exponential growth for 2d Euler near the boundaries, and other issues.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

偏微分方程 (PDE) 是科学和工程中使用的许多数学模型的基础。示例包括流体流动方程或描述各种结构中应力分布的方程。在实践中，通常使用计算机来求解方程，对方程的良好理论理解对于找到有效的算法非常重要。目前，我们对许多偏微分方程的理论理解还不完整。线性模型（我们对此有更好的理解）和非线性模型之间存在重要区别。在线性模型中，粗略地说，系统对扰动的反应与扰动成正比。在非线性模型中，情况并非如此，许多数学困难都可以追溯到这种效应。线性状态通常与平衡的小扰动相关，而大扰动通常由非线性现象控制。该项目将重点关注非线性现象。特别是，将要研究的方程组中最严重的影响之一是奇点的形成以及方程预测能力的相关损失。这将在基本方程（例如不可压缩流体力学方程）和各种模型方程的背景下进行研究，这可以为在困难的开放问题上取得进展提供合适的垫脚石。该项目为研究生提供研究培训机会。在更技术层面上，该项目重点关注以下领域：（i）展示流体力学偏微分方程特征的一维模型。其中包括 De Gregorio 模型（可以被视为 Constantin-Lax-Majda 模型的扩展）、二维系统边界行为建模方程以及向量值 Burgers 型方程。尽管它们很简单，但这些模型可以成为思想的良好来源，并且对它们的更好的理解可以导致基本方程的进展。事实上，在完整的三维不可压缩欧拉方程的背景下，回到这些模型的想法已经被证明是重要的； (ii) 物理和几何中出现的方程，我们有足够的机会获得相当好的理解。其中包括复杂的 Ginzburg-Landau 方程、二维调和图热流以及由向量值函数的多维变分积分产生的一些经典非线性椭圆系统。（最后一个主题与非线性弹性有关。）所有这些本身都是重要的方程，PI 相信与它们相关的一些长期悬而未决的问题可以成功解决； (iii) 不可压缩流体动力学完整方程（纳维-斯托克斯和欧拉方程）的可接近方面。这包括研究描述广义自相似奇点的方程的可解性、Leray-Hopf 解的可能非唯一性与不稳定性问题之间的关系、边界附近二维欧拉双指数增长的稳定性以及其他问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。