Three problems in fluid mechanics through the lens of sparseness of the regions of intense fluid activity

湍流无处不在，从客机后面的尾流湍流到太阳风湍流。尽管取得了重大进展，但对湍流运动的完全理解仍然是科学和工程领域的重大挑战之一。特别是，对相干涡旋结构（例如，翼尖涡旋或太阳风中的流片）在湍流中所起作用的精确描述仍然难以捉摸。强烈流体活动区域在这些连贯结构中自组织的事实也被称为湍流的“空间间歇性”。该项目的目标是利用一个新的数学框架来研究流体流动的空间间歇性，该框架是由首席研究员（PI）引入的，以解决该领域的几个开放问题，包括涡旋片卷起的经典问题。该项目还将为学生提供参与研究的机会。该项目的总体目标是展示基于强烈流体活动区域的空间间歇性的数学框架的更广泛的效用，该框架是由PI和合作者开发的，旨在弥合Navier-Stokes （NS）正则性问题中的尺度差距。该项目分为三个子项目。第一个是在超扩散的背景下。在这里，我们的目标是证明——只要拉普拉斯函数的幂严格大于1——非线性相互作用的局域性就意味着规律性。除了数学上的兴趣之外，从湍流物理学的角度来看，这也很有趣，因为非线性传递的局部性是湍流现象学的原则之一。第二个将考虑几个关键空间中的NS流，其中只有小数据规律性结果是已知的。这里的关键思想是通过根据局部方向极大函数在尺度上重新表述局部一维稀疏的概念来获得对数（在大数据的情况下）。第三个是经典的涡旋片卷问题。本文的主要目标是研究基于涡度超水平集的局部一维稀疏性的粘滞稳定效应。更准确地说，该框架将自然地识别出一系列约束条件，其中一边是粘度，另一边是薄片的厚度和曲率。该奖项反映了美国国家科学基金会的法定使命，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。

Turbulent flows are omnipresent, from the wake turbulence behind an airliner to the solar wind turbulence. Notwithstanding significant progress, a complete understanding of turbulent motion remains one of the grand challenges in science and engineering. In particular, a precise description of the role the coherent vortex structures (e.g., the wingtip vortices or the current sheets in the solar wind) play in turbulence remains elusive. The fact that the regions of the intense fluid activity self-organize in these coherent structures is also referred to as the 'spatial intermittency' of turbulent flows. The goal of the project is to utilize a novel mathematical framework for the study of spatial intermittency in fluid flows introduced by the principal investigator (PI) to tackle several open problems in the field, including the classical problem of the vortex sheet roll-up. The project will also provide opportunities for the involvement of the students in the research. The overarching objective of this project is to demonstrate the broader utility of the mathematical framework based on the spatial intermittency of the regions of intense fluid activity that had been developed by the PI and the collaborators in the quest to bridge the scaling gap in the Navier-Stokes (NS) regularity problem. The project branches into three subprojects. The first one is in the context of hyper-diffusion. Here, the goal is to show that–as soon as the power of the Laplacian is strictly greater than one–locality of the nonlinear interactions will imply regularity. In addition to the mathematical interest, this is also intriguing from the standpoint of the physics of turbulence since the locality of the nonlinear transfer is one of the tenets of turbulence phenomenology. The second one will consider the NS flows in several critical spaces in which only the small data regularity results are known. The key idea here is to gain a logarithm (in the case of large data) by reformulating the concept of local one-dimensional sparseness at scale in terms of the local directional maximal function. The third one concerns the classical problem of the vortex sheet roll-up. The main goal here is to investigate a stabilizing effect of the viscosity–considering the full NS system–based on local one-dimensional sparseness of the vorticity super-level sets. More precisely, the framework will naturally identify a family of constraints among viscosity, on one side, and the thickness and the curvature of the sheet on the other.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.