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.

部分微分方程（PDE）是基于科学和工程中使用的许多数学模型。示例包括流体流的方程或描述各种结构中应力分布的方程。实际上，方程通常是通过计算机使用来解决的，对方程式的良好理论理解对于查找有效算法很重要。目前，我们对许多PDE的理论理解是不完整的。线性模型（我们的理解更好）和非线性模型之间存在重要的区别。在线性模型中，系统对干扰的反应与干扰成正比。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果。线性体制通常与均衡的小扰动有关，而大型干扰通常受非线性现象的控制。该项目将集中于非线性现象。特别是，将要研究的方程式中最严重的效果之一是形成奇异性和方程式的相关预测能力损失。这将在基本方程式的背景下（例如不可压缩流体力学的方程式）以及各种模型方程式进行研究，这些方程式可以提供合适的垫脚石来在困难的开放问题上取得进展。该项目为研究生提供了研究培训机会。在更具技术水平的情况下，该项目着重于以下领域：（i）一维模型，展示了流体力学PDE的特征。其中包括De Gregorio模型（可以将其视为君士坦属 - 拉克斯-Majda模型的扩展），对2D系统的边界行为进行建模的方程和矢量值汉堡型方程。尽管它们的简单性，但这种模型还是可以成为思想的良好来源，并且他们的改善理解可以导致基本方程式的进步。实际上，在整个三维不可压缩的Euler方程的背景下，回到这些模型的想法已经被证明很重要。（ii）在物理和几何形状中产生的方程式，我们有一个令人满意的机会获得相当良好的理解。其中包括复杂的金茨堡 - 兰道方程，2D谐波热流量以及由矢量值函数的多维变分积分引起的一些经典的非线性椭圆系统。（最后一个主题与非线性弹性有联系。）所有这些都是重要的方程式，PI认为可以成功解决与它们有关的一些长期存在的开放问题；（iii）不可压缩流体动力学（Navier-Stokes和Euler方程）的完整方程式的平易近人方面。这包括对描述广义自相似奇点的方程式的可溶性的调查，可能的非唯一性之间的关系与不稳定性的问题与问题的问题之间的关系，两倍指数增长在边界附近的双重指数增长的稳定性以及其他问题的依据反映了NSF的法规及其范围，这是NSF的法规及其范围的影响。标准。