CAREER:Formation of Small Scales and Dissipation in Incompressible Fluids

流体动力学的应用范围从生物过程、空气动力学和气象学的研究，仍然是经典和现代物理学最重要的领域之一。尽管流体对几乎所有生命都很重要，但人们对如何预测一般环境中的流体运动知之甚少。这个项目的目标是推进我们对流体流动的一些基本问题的认识：经典物理学定律是否足以提供所有情况下流体流动的全貌？更确切地说，是否存在着支配流体运动的经典定律失效的极端事件？如果是这样的话，有没有一种方法可以修正经典定律，给出在这种情况下流体运动的有效模型？该奖项还支持博士后学者和研究生参与研究项目。在数学上，这些问题涉及到经典流体方程的全局可解性问题：欧拉方程和Navier-Stokes方程。这项工作的重点将是欧拉方程的奇异性形成的问题。我们将分析最近构造的奇点形成的例子的稳定性，并研究这些例子可以扩展到什么程度，以给出更规则的解决方案的奇点形成。同时，我们将考虑增强耗散的问题，并研究粘性问题中可能的奇异行为对粘性问题中快速耗散的影响。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

With applications ranging from the study of biological processes, aerodynamics, and meteorology, fluid dynamics remains one of the most important fields of classical and modern physics. Despite the importance of fluids to virtually all life, little is known about how to predict fluid motion in general settings. The goal of this project is to advance our knowledge of some of the fundamental questions regarding fluid flow: are the laws of classical physics sufficient to provide a complete picture of fluid flow in all scenarios? More precisely, are there extreme events in which the classical laws governing fluid motion break down? If so, is there a way to amend the classical laws to give effective models of fluid motion in such circumstances? This award also supports the involvement of a postdoctoral scholar and a graduate student in the research project. Mathematically, these questions relate to the question of the global solvability for the classical fluid equations: the Euler and Navier-Stokes equations. The focus of this work will be on the problem of singularity formation for the Euler equation. We will analyze the stability of recently constructed examples of singularity formation and study the extent to which these examples can be extended to give singularity formation for more regular solutions. In parallel, we will consider the problem of enhanced dissipation and study the effects of possible singular behavior in inviscid problems on rapid dissipation in viscous problems.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.