Pathological solutions of fluid equations

本项目研究有关流体运动方程的一些基本的公开问题。这些被物理学家和工程师广泛用于实际应用的方程是近两个世纪前提出的。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一概念的局限性。另一方面，该项目建议证明存在预期描述湍流的物理解。该项目还将为研究生参与该项目提供机会。在过去的几十年里，数学流体动力学已经突出了许多结构的解决方案的流体方程，表现出病态或野生的行为，如损失的能量平衡，非唯一性，奇点的形成，和耗散异常。有趣的是，从数学的角度来看，这些解决方案提供了反例，各种适定性的结果在超临界空间。此外，这些建设性的方法从物理角度来看也变得越来越相关。实际上，湍流的一个基本物理性质是能量级联的存在。由柯尔莫哥洛夫猜想，能量级联已经在实验和数值上观察到，但很难解析地产生。该项目的目的是使用最先进的方法，如凸积分和有效混合的新发展，不仅构建数学病理，而且物理现实的流体方程的解决方案。该奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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.