Singularities and mixing in Euler flows

现代科学中最古老和最耐人寻味的谜题之一就是揭开和理解我们与流体行为相关的不稳定和湍流的真实本质。因此，流体这个词可以指杯子中水和油的混合，或者更剧烈的过程，如恒星的坍塌。这两种看似不相关的物理情况之间有一条共同的数学线索，那就是著名的欧拉方程，早在1755年就由伦哈德·欧拉写下了。这是偏微分方程式最早的例子之一，至今仍是数学研究的焦点。该提议旨在严格研究描述奇异过程的欧拉方程的解，在该奇异过程中，运动的流体区域可以收缩到一点，扩展到无穷远，或者呈现出数学家喜欢称为拓扑奇点的某种其他形式。即使我们想描述发生在非常不同的空间尺度上的物理过程，但在精确的数学意义上，欧拉方程的解可以是“尺度无关的”，这一概念恰如其分地被称为自相似。所提议的研究的流A和B正好集中于寻找和研究欧拉方程的这种(近似)自相似解，因为它们是上述奇性的数学基石。在流A中，我们将重点放在描述正在收缩的恒星的引力崩塌这一著名问题上，而在流B中，我们将研究被流体包围的塌缩空穴的性质。在这两种情况下，数学分析、几何和物理之间令人惊讶的联系发挥了关键作用。在流C中，我们将注意力转向二维(或平面)流体流动，目的是加深我们对流体混合性质的了解。通过在咖啡中搅拌牛奶，人们可能会想，我们必须放大多远才能看到牛奶颗粒从咖啡颗粒中分离出来。事实证明，这个简单的问题激发了对混合的严格几何定义，这使得数学家能够提出猜想并证明定理。这里我们的重点是一个著名的问题，即布雷桑混合猜想，它与二维欧拉方程解的长期行为有关。

One of the oldest and most intriguing puzzles of modern science is to unravel and understand the true nature of instabilities and turbulence that we associate with the behaviour of fluids. Thereby the word fluid can refer to a mixing between between water and oil in a cup, or much more violent processes such as the collapse of a star. A common mathematical thread between these two seemingly unrelated physical situations is the famous Euler equation, written down by Leonhard Euler as early as 1755. This is among the earliest examples of partial differential equations, that to this day, remains a focal point of mathematical research. The proposal aims to rigorously examine solutions of Euler equations that describe singular processes within which a moving fluid region can shrink to a point, expand to infinity, or exhibit some other form of, what mathematicians like to call, a topological singularity.Even though we want to describe physical processes that happen at vastly different spatial scales, there is a precise mathematical sense in which the solutions of the Euler equation can be "scale-independent", a notion aptly termed self-similarity. Streams A and B of the proposed research focus precisely on finding and studying such (approximately) self-similar solutions of Euler equation as they are mathematical building blocks for the singularities mentioned above. In stream A we focus on the celebrated problem of describing the gravitational collapse of a shrinking star, while in stream B we examine the nature of collapsing cavities surrounded by a fluid. In both cases, surprising links between mathematical analysis, geometry, and physics play a key role.In stream C we turn our attention to 2-D (or planar) fluid flows, with the aim of deepening our knowledge of mixing properties of fluids. By stirring milk into the coffee, one may wonder how far must we zoom into the mixture before we see the milk particles separated from the coffee particles. This simple question, it turns out, motivates a rigorous geometric definition of mixing, which allows mathematicians to formulate conjectures and prove theorems. Our focus here is on a famous problem known as the Bressan's mixing conjecture and a link to the long-time behaviour of solutions to the 2-D Euler equation.