Topics in the Analysis of Nonlinear Partial Differential Equations

一些最有用的自然现象模型使用了经典场论和连续介质力学。在这些模型中，描述现实的对象是定义在时空区域中的满足某些偏微分方程的函数。国际和平研究所开展的研究集中在流体力学背景下产生的方程。偏微分方程解通常用无穷多个参数来描述。这是一个复杂的问题，但回报是，由此产生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出了一些非常合理的假设，我们就必然会得到不可压缩的Navier-Stokes方程。模型应该是什么基本上是没有歧义的，纳维尔-斯托克斯方程在科学和工程中得到了广泛的应用，从飞机设计、天气预报、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，有必要将连续介质模型简化为由有限多个参数描述的模型。然而，这一步并不规范，有许多合理的方法来做到这一点。当连续介质模型在数学上被很好地理解时，它对有限维模型的简化以及它们之间的关系也相对地被很好地理解。当然，人们不应该夸大这一点--即使在这种情况下，仍然有许多有趣的未决问题。然而，我们对Navier-Stokes方程的数学理解是相当不完整的，这使得对其有限维约化结果的解释变得更加困难。考虑这个项目中的研究的一种方式是，它旨在为填补我们知识中的这一空白做出贡献，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。在更技术性的层面上，本项目关注以下主题：(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方程及其相关方程的适定性和非唯一性的研究已经取得了重要进展，但重要的基本问题仍然悬而未决。例如，解算子能否从光滑解不断扩展到具有由能量等自然物理量生成的拓扑的空间？具有饱和边界的二维域上的正则性估计是否被某些解饱和？(Ii)三维Navier-Stokes方程的定态解和Serrin旋涡的变形。对稳态解的研究，除了具有独立的实际意义外，还为提高我们对依赖于时间的解的理解提供了见解。在这个项目中，重点是某些已知的解类(由于Serrin)具有对称性到具有较少对称性的解的变形。值得指出的是，Serrin的解和这里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。Liouville定理旨在描述所有时空中的整体有界解，并且与方程的公开正则性问题密切相关。在本项目中，它们主要是在稳态环境下进行研究的。他们的研究还导致将要解决的有趣的线性问题；(Iv)一维模型，包括研究通过四元数Burgers模型的一般强迫可能避免奇点的研究，以及de Gregorio模型的整体适定性和更详细的解行为的研究。即使在一维模型的水平上，也存在许多悬而未决的问题。这些问题应该为提高我们的理解并最终将从研究这些模型中学到的经验应用到更高的维度提供很好的垫脚石。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。

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.