权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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。例子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，方程通常使用计算机求解，对方程的良好理论理解对于找到有效的算法很重要。目前，我们对许多偏微分方程的理论认识是不完整的。线性模型（我们对此有更好的理解）和非线性模型之间有一个重要区别。在线性模型中，系统对扰动的反应大致上与扰动成比例。在非线性模型中，情况并非如此，许多数学困难可以追溯到这种效果。线性状态通常与来自平衡的小扰动有关，而大扰动通常由非线性现象控制。这个项目将侧重于非线性现象。特别是，一类方程中最严重的影响之一，将被调查的是形成奇点和相关损失的预测能力的方程。这将在基本方程（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适当的垫脚石。本项目为研究生提供研究培训机会。在更技术的层面上，本项目侧重于以下领域：（i）展示流体力学偏微分方程特征的一维模型。这些模型包括德格雷戈里奥模型（De Gregorio model）（可以被认为是康斯坦丁-拉克斯-马格达模型的扩展），二维系统的边界行为方程模型，以及向量值Burgers型方程。尽管这些模型很简单，但它们可以成为思想的良好来源，对它们的理解的提高可以导致基本方程的进展。事实上，回到这些模型的想法已经被证明是重要的，在全三维不可压缩欧拉方程的背景下;（二）方程产生的物理和几何，我们有一个令人满意的机会获得一个相当好的理解。这些包括复杂的Ginzburg-Landau方程，二维调和映射热流，和一些经典的非线性椭圆系统所产生的多维变分向量值函数。(The最后一个主题与非线性弹性有关。）所有这些都是重要的方程在自己的权利和PI认为，一些长期存在的开放问题，他们可以成功地解决;（iii）接近方面的完整方程的不可压缩流体动力学（Navier-Stokes方程和欧拉方程）。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，二维Euler方程在边界附近的双指数增长的稳定性，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的评估来支持。影响审查标准。