Topics in the Analysis of Nonlinear Partial Differential Equations

非线性偏微分方程分析专题

• 批准号: 2247027
• 负责人: Vladimir Sverak
• 金额: $ 58.29万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247027&HistoricalAwards=false
• 关键词: Topics; Analysis; Nonlinear; Partial; Differential

Some of the most useful models of natural phenomena use classical field theories and continuum mechanics. In these models, the objects aiming to describe reality are functions defined in regions of space-time and satisfying certain partial differential equations. The research carried out by the PI focuses on the equations arising in the context of fluid mechanics. The solutions of partial differential equations are usually described by an infinite number of parameters. This is a complication, but the reward is that the resulting models are in some sense canonical. For example, once we make a small number of very reasonable assumptions about fluid motion, we necessarily end up with the incompressible Navier-Stokes equation. There is essentially no ambiguity about what the model should be, and the Navier-Stokes equation is widely used in science and engineering, from airplane design, weather prediction, and climate modeling to computations of various water flows. To be able to perform computer simulations of the equations, it is necessary to reduce the continuum models to models described by finitely many parameters. However, this step is not canonical, there are many reasonable ways of doing it. When the continuum model is well-understood mathematically, its reductions to finite-dimensional models and their relations are relatively well understood. Of course, one should not overstate this - even in that case there are still many interesting open problems. However, our mathematical understanding of the Navier-Stokes equation is quite incomplete, and that makes the interpretations of the results from its finite-dimensional reductions harder. One way to think about the research in this project is that it aims to contribute to filling this gap in our knowledge and ultimately make our modeling more efficient. The project provides research training opportunities for graduate students.At a more technical level, this project focuses on the following topics: (i) Well-posedness, (non)-uniqueness, backward uniqueness, and critical situations for basic equations. In recent years, important advances have been made concerning well-posedness and non-uniqueness for the Navier-Stokes and related equations, but important basic questions still remain open. For example, can the solution operator be continuously extended from smooth solutions to spaces with topologies generated by natural physical quantities such as energy? Are regularity estimates in two-dimensional domains with boundary saturated by some solutions? (ii) Steady-state solutions of the three-dimensional Navier-Stokes equation and deformations of Serrin’s swirling vortex. The study of steady-state solutions, in addition to being of independent practical interest, provides insights for improving our understanding of time-dependent solutions. In this project, the focus is on deformations of certain known classes of solutions (due to Serrin) with symmetries to solutions with fewer symmetries. It is worth pointing out that both Serrin's solutions and the deformations envisaged here have connections to tornadoes; (iii) Liouville Theorems and related linear problems. Liouville theorems aim to describe global bounded solutions in all space-time and are closely related to open regularity questions about the equation. In this project, they are studied mostly in the steady-state setting. Their study also leads to interesting linear problems that will be addressed; (iv) One-dimensional models, including the study of possible avoidance of singularities by generic forcing for the quaternionic Burgers model and the global well-posedness and more detailed solution behavior for the De Gregorio model. There are many open problems even at the level of the one-dimensional models. These problems should provide good steppingstones towards improving our understanding and eventually applying the lessons learned from studying these models to higher dimensions.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.

一些最有用的自然现象模型使用经典场论和连续介质力学，在这些模型中，旨在描述现实的对象是在时空区域中定义并满足某些偏微分方程的函数。专注于流体力学背景下的方程。偏微分方程的解通常由无限多个参数来描述，但这样做的好处是所得模型在某种意义上是规范的。一旦我们做了少量关于流体运动的非常合理的假设，我们必然会得到不可压缩的纳维-斯托克斯方程。模型应该是什么基本上没有任何歧义，并且纳维-斯托克斯方程广泛应用于科学和工程领域，从飞机设计，天气预测和气候建模到各种水流的计算为了能够​​对方程进行计算机模拟，有必要将连续模型简化为由有限多个参数描述的模型。当连续统模型在数学上被很好地理解时，它是规范的，有许多合理的方法可以很好地理解它对有限维模型的简化以及它们之间的关系当然，人们不应该夸大这一点 - 即使在这种情况下也是如此。然而，我们对纳维-斯托克斯方程的数学理解还相当不完整，这使得解释其有限维约简结果变得更加困难。它旨在帮助填补我们的知识空白，并最终使我们的建模更加高效。该项目为研究生提供研究培训机会。在更技术的层面上，该项目重点关注以下主题：（i）适定性，基本方程的（非）唯一性、后向唯一性和临界情况 近年来，在纳维-斯托克斯及相关方程的适定性和非唯一性方面取得了重要进展，但重要的基本问题仍然悬而未决。例如，解决方案可以算子可以从平滑解连续扩展到具有由自然物理量（例如能量）生成的拓扑的空间吗？具有边界的二维域中的正则性估计是否被某些解饱和？（ii）三维纳维-斯托克斯的稳态解？ Serrin 旋转涡流的方程和变形 对稳态解的研究除了具有独立的实际意义外，还为提高我们对瞬态解的理解提供了见解。在该项目中，重点是某些已知类别的变形。具有对称性的解（由于 Serrin）到具有较少对称性的解 值得指出的是，Serrin 的解和此处设想的变形都与龙卷风有关；（iii）刘维尔定理和相关的线性问题。所有时空中的有界解，并且与方程的开放正则性问题密切相关。在这个项目中，它们主要在稳态设置下进行研究，它们的研究也导致了有趣的线性问题。将解决；（iv）一维模型，包括通过四元数 Burgers 模型的通用强迫和 De Gregorio 模型的全局适定性和更详细的解行为来研究可能避免奇点的问题。即使在一维模型的层面上，这些问题也应该为提高我们的理解并最终将研究这些模型中学到的经验教训应用到更高的维度提供良好的基础。该奖项的法定使命是通过使用基金会的智力价值和更广泛的影响审查标准。