STICKS - Stochastic Mikado Flows for Fluid Mechanics

STICKS - 流体力学的随机 Mikado 流

• 批准号: 507913792
• 负责人: Dr. Andre Schenke
• 金额: --
• 依托单位: Courant Institute of Mathematical Sciences
• 依托单位国家: 德国
• 项目类别: WBP Fellowship
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/507913792?language=en
• 关键词: STICKS; Stochastic; Mikado; Flows; Fluid

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.

流体动力学和湍流在科学技术中扮演着极其重要的角色，应用范围从工程学到气候学和天体物理学。模拟流体流动的偏微分方程组(PDE)产生了几个自相矛盾的弱解。例如，弱解决方案可能在完全休息时开始，然后突然开始以最不规律的方式移动，最后又完全安静下来。这种不规则的行为显然违反了能量的基本守恒定律，并被认为是湍流形成的重要原因，在湍流中，它被称为反常耗散。在过去的20年里，它一直是密集研究的主题，最近导致了对所谓的Onsager猜想的严格数学证明，该猜想准确地描述了一种流动必须有多平滑才能保持能量。这个猜想的证明使用了一个更古老的理论，称为凸积分，由诺贝尔和阿贝尔奖获得者约翰·F·纳什和阿贝尔奖获得者米哈尔·L·格罗莫夫首创。最近，这些技术被几个小组应用于随机偏微分方程，其流动受到数值、经验和物理的不确定性的影响，最近在马丁·海尔的菲尔兹奖章之后变得非常流行，其中凸积分导致了与确定性情形类似的壮观结果。在这个项目中，我们将从流体力学和等离子体物理中研究几个随机PDE模型，并通过凸积分理论研究它们的反常耗散性质。更确切地说，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(MHD)方程。该项目将在纽约库兰特数学科学研究所的Vlad Vicol教授的小组中完成。我们将发展随机Mikado流的数学理论作为凸积分中的关键构件，并将其应用于上述系统，目的是得到这些方程解的非唯一性结果。