CAREER: Singularities in fluids

无数的流体动力学现象都是用奇点来描述的，其中最广为人知的可能是在空气或水等运动速度超过局部音速的介质中受到扰动时所产生的冲击波。该项目的目标是围绕流体中奇点的研究，为研究和教学活动创建一个广泛的计划。该奖项将为博士后学者提供研究机会和培训，并将利用其研究要素设计适合本科生的项目。该项目还将致力于编写一本关于可压缩流体中冲击波的基础研究生教科书。研究部分分为三个项目：冲击波的形成和发展、平滑初始数据的径向内爆和基于神经网络的自相似爆破。在PI及其合作者以前工作的基础上，第一个项目的目的将是为多维可压缩欧拉方程提供第一个关于激波形成和发展的完整描述。第二个项目涉及进一步发展关于等熵可压缩流动的自相似内爆解的先前工作，以调查新类型奇点的可能性。第三个项目涉及利用物理上知情的神经网络来搜索流体中新形式的奇点，这种奇点的存在将通过计算机辅助证明的帮助而变得严格。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Innumerable hydrodynamical phenomena are described in terms of singularities, of which perhaps the most well-known is the formation of shock waves resulting from a disturbance in a medium such as air or water moving faster than the local speed of sound. The goal of this project is to create a broad program for both research and pedagogical activities centered around the study of singularities in fluids. The award will provide research opportunities and training for postdoctoral scholars and will leverage its research elements to design projects suitable for undergraduate students. The project will also aim at producing a foundational graduate textbook on shock waves in compressible fluids.The research component is split into three projects: formation and development of shock waves, radial implosions from smooth initial data, and self-similar blow-up via neural networks. Building on previous work of the PI and his collaborators, the aim of the first project will be to provide the first full description of shock wave formation and development for the multi-dimensional compressible Euler equations. The second project involves further developing prior work on self-similar imploding solutions for isentropic compressible flows to investigate the possibility of new types of singularities. The third project involves utilizing physically informed neural networks to search for new forms of singularities in fluid, whose existence will be made rigorous through the aid of computer assisted proofs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.