STICKS - Stochastic Mikado Flows for Fluid Mechanics

流体动力学和湍流在科学和技术中发挥着极其重要的作用，其应用范围从工程到气候学和天体物理学。模拟流体流动的偏微分方程（PDEs）产生了几个矛盾的弱解。例如，一个弱溶液可能从完全静止开始，然后突然开始以最不规则的方式运动，最后又完全静止下来。这种不规则的行为显然违反了基本的能量守恒，并且被推测在湍流的形成中是重要的，它被称为反常耗散。在过去的二十年里，它一直是一个密集研究的主题，最近导致了所谓的Onsager猜想的数学严谨证明，该猜想精确地描述了为了保持能量，流动必须有多平滑。这个猜想的证明使用了一个更古老的理论，叫做凸积分，由诺贝尔奖和阿贝尔奖得主约翰·f·纳什和阿贝尔奖得主米哈伊尔·l·格罗莫夫率先提出。最近，这些技术已经被几个小组应用到随机微分方程中，这些微分方程模拟了受数值、经验和物理不确定性影响的流动，最近在马丁·海勒的菲尔兹奖之后变得非常流行，其中凸积分导致了与确定性情况类似的惊人结果。在本项目中，我们将研究流体力学和等离子体物理学中的几种随机PDE模型，并通过凸积分理论研究它们的异常耗散特性。更准确地说，我们将研究随机扰动下的输运方程、欧拉方程和磁流体动力学方程。该项目将在纽约Courant数学科学研究所的Vlad Vicol教授小组进行。我们将发展随机Mikado流的数学理论作为凸积分的关键构建块，并将其应用于上述系统，旨在得到这些方程解的非唯一性结果。

Fluid dynamics and turbulence play an extraordinarily important role in science and technology, with applications ranging from engineering to climatology and astrophysics. The partial differential equations (PDEs) that model the flow of fluids give rise to several paradoxical, weak solutions. For example, a weak solution could start at a complete rest, then suddenly start to move in a most irregular fashion, only to fall completely quiet again in the end. This kind of irregular behaviour clearly violates the fundamental conservation of energy and is conjectured to be important in the formation of turbulence, where it is known as anomalous dissipation. It has been the subject of intense research in the last two decades, leading recently to a mathematically rigorous proof of the so-called Onsager conjecture that gives a precise description of how smooth a flow has to be in order to be energy-preserving. The proof of this conjecture used a much older theory called convex integration, pioneered by Nobel and Abel laureate John F. Nash and Abel laureate Mikhael L. Gromov.More recently, these techniques have been applied by several groups to stochastic PDEs which model flows subjected to numerical, empirical and physical uncertainties, and which have recently become very popular in the wake of Martin Hairer's Fields Medal, where convex integration led to similarly spectacular results as in the deterministic case.In this project, we will investigate several stochastic PDE models from fluid dynamics and plasma physics and study their anomalous dissipation properties via the theory of convex integration. More precisely, we will study transport equations, Euler equations and magnetohydrodynamic (MHD) equations, subject to random perturbations.The project will be spent in Prof. Vlad Vicol's group at Courant Institute of Mathematical Sciences, New York. We will develop a mathematical theory of stochastic Mikado flows as key building blocks in convex integration and apply it to the above-mentioned systems, aiming to get nonuniqueness results for solutions to these equations.