Collaborative research: Turbulent cascades and regularity theory in physical scales of 3D incompressible fluid flows

Grujic，DMS-1212023 Dascaliuc，DMS-1211413 研究人员的目标是推进我们对湍流现象的理解，并提供新的见解，在三维粘性不可压缩流体中由三维Navier-Stokes方程（NSE）描述的可能的奇异性形成的问题。 该项目扩展了他们最近关于能量级联的存在和位置以及物理尺度上的异常耗散的结果。 主要目标是，一方面，提出Onsager临界Morrey型条件，防止3D不可压缩流中的异常耗散，并发现一组通用的标度律-让人想起基本的Kolmogorov K41定律-自然地从他们的物理尺度湍流级联理论中散发出来，另一方面，提出数学证据的几何方案关闭缩放差距的规律性问题，即，的关键性的3D NSE问题的大数据。 其关键思想是使用一种新的整体平均过程，能够在给定的尺度上检测感兴趣的物理密度的显着符号波动。 这种方法在空间上是高度局部的，并且适用于不满足均匀性和各向同性假设的流动。 此外，它提供了一个框架的研究几何相干结构的物理空间。 2000年，克莱数学研究所确定了数学中的七个“千年问题”，这些问题的解决方案将对21世纪的社会产生广泛的影响。 这些问题之一是提供一个严格的数学基础的任务（即，解决奇点形成的可能性）的Navier-Stokes方程描述的流体运动。 奇点的存在性是任何现实生活中现象的数学建模的基本问题。 在Navier-Stokes方程的情况下，这个规律性问题被认为与湍流的概念有关。 尽管用于研究湍流的经验建模和计算技术取得了成功，但这种现象仍然是现代物理学尚未解决的基本问题之一。 在理解湍流方面的任何可衡量的进展都可能在天气和气候建模以及工程应用等领域产生广泛的影响。 一方面，抑制湍流对于设计和工程更高效的车辆、风力涡轮机、管道系统至关重要。 另一方面，在各种纳米级工程设计以及生物医学工程中，都需要增强湍流混合-一个例子是更有效的药物输送系统。 该项目有助于促进了解如何更好地预测和检测湍流行为，以及可能的控制它。研究人员从事研究生和本科生的项目，并提出讲座湍流当地高中生。

Grujic, DMS-1212023Dascaliuc, DMS-1211413 The investigators aim to advance our understanding of the phenomenon of turbulence, as well as to offer new insights into the problem of possible singularity formations in 3D viscous incompressible fluids described by the 3D Navier-Stokes equations (NSE). The project extends their recent results regarding existence and locality of energy cascades, and anomalous dissipation in physical scales of the flow. The main goals are, on one hand, to present Onsager-critical Morrey-type conditions preventing anomalous dissipation in 3D incompressible flows and to discover a set of universal scaling laws -- reminiscent of the fundamental Kolmogorov K41 laws -- that naturally transpire from their theory of turbulent cascades in physical scales, and on the other hand, to present mathematical evidence of a geometric scenario closing the scaling gap in the regularity problem, that is, of the criticality of the 3D NSE problem for large data. The key idea is to use a new ensemble-averaging process capable of detecting significant sign fluctuations of the physical density of interest at the given scale. This approach is highly local in space and is applicable to flows that do not satisfy homogeneity and isotropy assumptions. In addition, it provides a framework for the study of geometrically coherent structures in physical space. In 2000, the Clay Mathematical Institute identified seven "Millennium Problems" in mathematics, solutions to which would have a broad impact on society in the 21st century. One of these problems is the task of providing a rigorous mathematical foundation (namely, addressing the possibilities of singularity formations) for the Navier-Stokes equations that describe the motion of fluids. Existence of singularities is a fundamental issue for mathematical modeling of any real-life phenomenon. In the case of Navier-Stokes equations this regularity problem is believed to be linked with the notion of turbulence. Despite successes in empirical modeling and computational techniques used to study turbulence, this phenomenon remains one of the unresolved fundamental problems of modern physics. Any measurable progress in understanding turbulence can have a broad impact in such areas as weather and climate modeling and engineering applications. On one hand, suppressing turbulence is essential in designing and engineering more efficient vehicles, wind turbines, pipeline systems. On the other hand, enhancing turbulent mixing is desired in a variety of nano-scale engineering designs, as well as in biomedical engineering -- one example being more efficient drug delivery systems. The project helps advance understanding of how to better predict and detect turbulent behavior, as well as possibly control it. The investigators engage graduate and undergraduate students in the project and present lectures on turbulence to local high school students.