Pathological solutions of fluid equations

这个项目研究了关于流体运动方程的一些基本的开放问题。这些方程式在近两个世纪前被物理学家和工程师广泛用于实际应用。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，弱解的概念被数学家广泛使用，它的存在是已知的。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有望描述湍流的物理解。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，数学流体动力学通过构造大量表现出病态或狂野行为的流体方程的解来突出，如能量平衡的丧失、非唯一性、奇异性的形成和耗散异常。从数学的角度来看，这些解很有趣，它们为超临界空间中的各种适定性结果提供了反例。此外，从实际角度来看，这些建设性的方法也变得越来越重要。事实上，湍流的一个基本物理性质就是能量级联的存在。科尔莫戈洛夫猜想，能量级联已经在实验和数值上观察到了，但很难用解析的方法来产生。该项目的目的是使用最先进的方法，如凸积分和高效混合方面的新发展，不仅构建流体方程的数学病态解，而且构建物理上现实的流体方程解。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。

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.