Anomalous dissipation in fluids, deterministic turbulence, and intermittency

1210896流体动力学中出现的最紧迫的数学问题正日益与湍流统计理论交织在一起。Navier-Stokes方程和Euler方程已经成为数值模拟的主要工具，但它们是否能实际产生满足湍流定律的解的最基本问题尚未得到解决。近年来，随着拓扑学方法的进步和对方程的非线性结构的更深入的了解，这个问题，俗称Onsager猜想，已经变得触手可及。这个项目的基本主题是开发一种分析方法，用于研究确定性解决方案的统计性质。重点是在Onsager临界正则性类中寻找Euler方程和活动标量方程的定常能量耗散弱解，对作为Richardson级联非均匀体积度量的间歇的经验概念进行严格的分形分析，排除大量实验数据所显示的对经典Kolmogorov定律的极端偏离，并在间歇与整体正则性问题之间建立联系。湍流是一种复杂的流体运动的混乱过程，通常发生在搅拌力向系统注入大量能量的情况下，比如来自太阳的热量或快速穿过空气的飞机。这个运动变得如此复杂和粗糙，以至于一部分动能丢失，产生了所谓的反常耗散。在我们的生活中，我们可以看到这种消散负责，例如，在飞机或汽车的湍流尾迹中产生额外的阻力。因此，为了提高能源效率，必须了解异常能量损失过程背后的机制及其产生的原因。在这个项目中，研究人员直接基于流体运动的控制方程，建立了湍流这一方面的数学基础。特别强调研究能量耗散的不均匀性，它可以识别出能量耗散最多的区域。该项目的一个组成部分是培养研究生和本科生的分析和计算技能。

ShvydkoyDMS-1210896 The most pressing mathematical issues arising in fluid dynamics are becoming increasingly intertwined with the statistical theory of turbulence. The Navier-Stokes and Euler equations have served as the primary tools for numerical modelling, yet the very basic question of whether they can actually produce solutions satisfying laws of turbulence has not been settled. In recent years, with the advances of topological methods and better understanding of the nonlinear structure of the equations, this problem, commonly known as the Onsager conjecture, has come within our reach. The underlying theme of this project is to develop an analytical approach for studying statistical properties of deterministic solutions. The focus is on finding stationary energy dissipative weak solutions of the Euler and active scalar equations in the Onsager-critical regularity class, developing rigorous fractal analysis of the empirical concept of intermittency as a volumetric measure of non-uniformity of the Richardson cascade, excluding extreme deviations from the classical Kolmogorov laws as shown by the vast experimental data, and establishing links between intermittency and the global regularity problem. Turbulence is a complex chaotic process of fluid motion that commonly happens when a stirring force, like heat from the sun or an airplane cutting fast through air, injects a lot of energy into the system. This motion gets so complicated and rough that a part of the kinetic energy gets lost, creating what is known as anomalous dissipation. In our lives we can see this dissipation responsible, for example, for creation of additional drag in the turbulent wake of an airplane or our cars. In order to improve energy efficiency it is therefore essential to understand the mechanisms behind the process of anomalous energy loss and the reasons why it arises. In this project the investigator develops mathematical foundations of this aspect of turbulence based directly on the governing equations of fluid motion. A particular emphasis is given to studying non-uniformity of energy dissipation, which allows identifying regions where it occurs the most. An integral part of the project is developing the analytical and computational skills of graduate and undergraduate students.