Singularities and mixing in Euler flows

现代科学中最古老、最有趣的难题之一，就是解开和理解与流体行为有关的不稳定性和湍流的真正本质。因此，“流体”这个词可以指杯子里的水和油的混合，或者更剧烈的过程，比如恒星的坍缩。这两种看似无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，它是由莱昂哈德·欧拉早在1755年写下的。这是偏微分方程最早的例子之一，直到今天，偏微分方程仍然是数学研究的焦点。该提案旨在严格检查欧拉方程的解，欧拉方程描述了奇异过程，其中移动的流体区域可以收缩到一点，扩展到无穷大，或者表现出数学家喜欢称之为拓扑奇点的其他形式。尽管我们想要描述发生在不同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”，这一概念被恰当地称为自相似性。本研究的流A和流B恰恰集中于寻找和研究欧拉方程的这种（近似）自相似解，因为它们是上述奇点的数学构建块。在流A中，我们专注于描述一颗正在收缩的恒星的引力坍缩的著名问题，而在流B中，我们研究了被流体包围的坍缩腔的性质。在这两种情况下，数学分析、几何和物理之间的惊人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特性的了解。把牛奶搅拌到咖啡里，人们可能会想，我们必须把混合物放大多远，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。我们这里的重点是一个著名的问题，即布雷桑混合猜想，以及与二维欧拉方程解的长期行为的联系。

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.