Pathological solutions of fluid equations

本课题研究流体运动方程的一些基本开放性问题。物理学家和工程师在现实生活中广泛使用的这些方程是在近两个世纪前引入的。然而，一些基本问题，如经典解的存在性和唯一性，仍然没有得到回答。因此，弱解的概念，其存在性是已知的，已被数学家广泛使用。该项目将通过构造具有各种非物理性质的弱解来证明这种概念的局限性。另一方面，该项目提出要证明物理解的存在，这些解有望描述湍流。这个项目也将为研究生提供参与项目的机会。在过去的几十年里，数学流体动力学已经被许多表现出病态或狂野行为的流体方程的解的构造所突出，例如能量平衡的丧失、非唯一性、奇点形成和耗散异常。有趣的是，从数学的角度来看，这些解提供了超临界空间中各种适定性结果的反例。此外，从实际的角度来看，这些建设性的方法也变得越来越相关。事实上，湍流的一个基本物理性质就是能量级联的存在。由Kolmogorov推测，能量级联已经在实验和数值上被观察到，但很难用解析方法产生。本项目的目的是利用最先进的方法，如凸积分和高效混合的新发展，不仅在数学上病态地，而且在物理上真实地构建流体方程的解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project studies some fundamental open questions concerning the equations of fluid motion. The equations, widely used by physicists and engineers for real-life applications, were introduced almost two centuries ago. However, some essential questions, such as the existence and uniqueness of classical solutions, are still not answered. Therefore, a notion of weak solutions, whose existence is known, has been extensively used by mathematicians. The project will demonstrate a limitation of this notion by constructing weak solutions with various unphysical properties. On the other hand, the project proposes to prove the existence of physical solutions that are expected to describe turbulent flows. This project will also provide opportunities for the involvement of graduate students in the project. In the past couple of decades, mathematical fluid dynamics has been highlighted by numerous constructions of solutions to fluid equations that exhibit pathological or wild behavior, such as loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, these solutions provide counterexamples to various well-posedness results in supercritical spaces. Moreover, these constructive approaches are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, the energy cascade has been observed both experimentally and numerically but had been difficult to produce analytically. The purpose of this project is to use state-of-the-art methods, such as convex integration and new developments in efficient mixing, to construct not only mathematically pathological, but also physically realistic solutions of fluid equations.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.