流体流动的概率描述：综述
Probabilistic Descriptions of Fluid Flow: A Survey
AI总结：
该综述针对确定性弱解的局限性，梳理了流体方程概率性解的两类主要框架（测度值与统计框架）及二者关联，聚焦可压缩与不可压缩Euler方程，为相关研究提供文献参考。
摘要：
流体的运动状态可能呈现高度无规则的湍流形态。人们早已认识到，在研究流体动力学的基本偏微分方程（即可压缩与不可压缩Navier-Stokes方程或Euler方程）时，需要引入某种弱解概念。标准的弱解概念（分布意义下的弱解）仍属于确定性范畴，因为它能给出几乎所有时空点上状态变量（如速度或密度）的精确值。然而，观测结果与数学理论均表明，这种确定性视角存在一定局限性。因此，近年来数学流体学界对概率性解概念的研究兴趣日益增长。由于这类概念数量繁多，梳理相关的经典与最新文献已颇具难度。本文旨在对流体方程的概率性公式表述进行较为简明却相当详尽的综述，这类表述大致可分为测度值框架与统计框架两类。我们讨论了这两种方法及其相互关系，以及各类统计公式表述之间的内在联系，研究重点为可压缩与不可压缩Euler方程。
英文摘要：
Fluids can behave in a highly irregular, turbulent way. It has long been realised that, therefore, some weak notion of solution is required when studying the fundamental partial differential equations of fluid dynamics, such as the compressible or incompressible Navier-Stokes or Euler equations. The standard concept of weak solution (in the sense of distributions) is still a deterministic one, as it gives exact values for the state variables (like velocity or density) for almost every point in time and space. However, observations and mathematical theory alike suggest that this deterministic viewpoint has certain limitations. Thus, there has been an increased recent interest in the mathematical fluids community in probabilistic concepts of solution. Due to the considerable number of such concepts, it has become challenging to navigate the corresponding literature, both classical and recent. We aim here to give a reasonably concise yet fairly detailed overview of probabilistic formulations of fluid equations, which can roughly be split into measure-valued and statistical frameworks. We discuss both approaches and their relationship, as well as the interrelations between various statistical formulations, focusing on the compressible and incompressible Euler equations.
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