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

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。例如用于流体流动的方程或描述各种结构中的应力分布的方程。在实践中，这些方程通常是用计算机来求解的，对这些方程有一个很好的理论理解对于找到有效的算法是很重要的。目前，我们对许多偏微分方程组的理论认识还不完全。线性模型(我们对线性模型的理解更好)和非线性模型之间有一个重要的区别。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影响。线性系统通常与来自平衡点的小扰动有关，而大扰动通常受非线性现象的支配。本项目将重点研究非线性现象。特别是，在将要研究的这类方程中，最严重的影响之一是奇点的形成和方程预测能力的相关损失。这将在基本方程(如不可压缩流体力学的方程)和各种模型方程的背景下进行研究，这些方程可以为在困难的开放问题上取得进展提供适当的垫脚石。该项目为研究生提供了研究培训的机会。在更高的技术层面上，该项目侧重于以下领域：(I)展示流体力学偏微分方程组特征的一维模型。其中包括de Gregorio模型(可以被认为是Constantin-Lax-Majda模型的扩展)，模拟2D系统边界行为的方程，以及矢量值Burgers型方程。尽管它们很简单，但这样的模型可能是一个很好的想法来源，它们更好的理解可以导致在基本方程上的进展。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)在物理和几何中出现的方程，我们有令人满意的机会获得相当好的理解。其中包括复Ginzburg-Landau方程，二维调和映射热流，以及由向量值函数的多维变分积分产生的一些经典的非线性椭圆型方程组。(最后一个主题与非线性弹性有关。)所有这些方程本身都是重要的，国际和平组织相信，与它们相关的一些长期悬而未决的问题可以被成功地解决；(Iii)不可压缩流体动力学完整方程(纳维尔-斯托克斯方程和欧拉方程)的可接近的方面。这包括研究描述广义自相似奇点的方程的可解性，Leray-Hopf解的可能的非唯一性与关于不稳定性的问题之间的关系，边界附近二维Euler的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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.