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定义在时空区域中并满足某些偏微分方程的函数。PI开展的研究侧重于流体力学背景下产生的方程。偏微分方程的解通常由无穷多个参数来描述。这是一个复杂的问题，但回报是所得到的模型在某种意义上是规范的。例如，一旦我们对流体运动做了少量非常合理的假设，我们就必然会得到不可压缩的纳维尔-斯托克斯方程。关于模型应该是什么基本上没有歧义，纳维-斯托克斯方程广泛用于科学和工程，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许多参数描述的模型。然而，这一步并不是规范的，有很多合理的方法可以做到这一点。当连续统模型在数学上得到很好的理解时，它到有限维模型的约化及其关系就相对得到了很好的理解。当然，人们不应该夸大这一点-即使在这种情况下，仍然有许多有趣的开放问题。然而，我们对Navier-Stokes方程的数学理解是相当不完整的，这使得从其有限维简化的结果的解释更加困难。考虑这个项目中的研究的一种方式是，它旨在填补我们知识中的这一空白，并最终使我们的建模更有效。该项目为研究生提供了研究培训的机会。在更技术的层面上，该项目侧重于以下主题：（i）适定性，（非）唯一性，向后唯一性，和基本方程的临界情况。近年来，关于Navier-Stokes方程及相关方程的适定性和非唯一性的研究取得了重要进展，但重要的基本问题仍然没有解决。例如，解算子能否从光滑解连续地扩展到由自然物理量（如能量）生成的拓扑空间？边界饱和的二维区域的正则性估计是否为某些解所饱和？(ii)三维Navier-Stokes方程的定常解和Serrin涡旋的变形。稳态解的研究除了具有独立的实际意义外，还为提高我们对时间依赖解的理解提供了见解。在这个项目中，重点是变形的某些已知类别的解决方案（由于塞林）与对称性的解决方案与较少的对称性。值得指出的是，塞林的解决方案和变形设想在这里有连接到龙卷风;（三）刘维尔定理和相关的线性问题。刘维尔定理旨在描述所有时空中的整体有界解，并且与方程的开放正则性问题密切相关。在这个项目中，他们主要是在稳态设置的研究。他们的研究还导致有趣的线性问题，将得到解决;（四）一维模型，包括研究可能避免奇异性的通用迫使四元数的Burgers模型和全球适定性和更详细的解决方案行为的德格雷戈里奥模型。即使在一维模型的水平上，也有许多悬而未决的问题。这些问题应该提供很好的垫脚石，以提高我们的理解，并最终应用从研究这些模型中获得的经验教训，以更高的dimensions.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。