Finite time blowup for supercritical equations, and correlations of multiplicative functions

本项目涉及两个数学研究方向。第一个方向涉及波的数学研究，如水波或声波。物理实验和数值模拟都显示了湍流现象：最初非常平稳流动和缓慢变化的流体可以发展成更复杂的漩涡和精细尺度的行为。目前尚不清楚模拟流体动力学的方程是否会导致“爆炸”，即模拟流体的某些部分达到无限大速度。这个项目研究的是，通过调整流体力学模型的方程式，比如改变空间的维数，可以在多大程度上“设计”爆炸。第二个方向涉及与数论中臭名昭著的孪生素数猜想有关的问题。这个猜想断言有无限多对素数，它们之间的距离为2，例如3和5、5和7、11和13、17和19，等等。这一论断是否正确尚不清楚，但最近在理解更容易处理的问题方面取得了很大进展，例如，相隔2的一对数都有奇数个质因数的情况发生的频率是多少。这个项目的目标是进一步发展这些有前途的新技术，在证明或反驳孪生素数猜想和数论中其他困难的、重要的问题上有潜在的应用。更详细地说，该项目的流体动力学部分主要关注不可压缩流体的欧拉方程的变体，特别是黎曼流形上的高维欧拉方程。选择这种流形的度规的能力提供了一种很有前途的方法来“编程”方程以显示某些理想的行为。先前的工作已经确定，某些二次常微分方程的动力学可以被编程到这样的系统中。该项目将继续致力于展示这些模型的有限时间膨胀（或其他有趣的行为，如图灵通用性）。对于该项目的数论方面，研究者和合作者的目标是通过将最近开发的熵减量方法与解析数论、组合学和遍历理论的技术相结合，建立关于Liouville函数相关性的Chowla猜想（或其对数平均变体）的进一步案例。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 concerns two mathematical research directions. The first direction involves the mathematical study of waves such as water waves or sound waves. Both physical experiments and numerical simulations exhibit the phenomenon of turbulence: fluids that are initially very smoothly-flowing and slowly-varying can develop much more complicated eddies and fine-scale behavior. It remains unknown whether the equations that model fluid dynamics can give rise to "blow-up," in which some portion of the simulated fluid achieves infinite velocity. This project studies the extent to which blow-up can be "engineered" by tweaking the equations that model fluid mechanics, such as changing the number of dimensions in space. The second direction involves questions related to the notorious twin prime conjecture in number theory. This conjecture asserts that there are infinitely many pairs of prime numbers that are separated by a distance of two, such as 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. It is still not known whether the assertion is true, but recently much progress has been made on understanding more tractable questions, such as how often it occurs that a pair of numbers, separated by a distance of two, both have an odd number of prime factors. This project aims to develop these promising new techniques further, with potential application to proving or disproving the twin primes conjecture and to other difficult, important questions in number theory. In more detail, the fluid dynamical part of the project focuses primarily on variants of the Euler equations for incompressible fluids, especially higher-dimensional Euler equations on Riemannian manifolds. The ability to select the metric of such a manifold gives a promising way to "program" the equations to exhibit certain desirable behavior. Prior work has established that the dynamics of certain quadratic ordinary differential equations can be programmed into such systems. The project will continue work towards exhibiting finite-time blow-up (or other interesting behavior, such as Turing universality) for these models. For the number-theoretic aspects of the project, the investigator and collaborators aim to establish further cases of the Chowla conjecture (or its logarithmically-averaged variants) on correlations of the Liouville function, by combining the recently-developed entropy decrement method with techniques from analytic number theory, combinatorics, and ergodic theory.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.