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

超临界方程的有限时间爆炸以及乘法函数的相关性

• 批准号: 1764034
• 负责人: Terence Tao
• 金额: $ 68.05万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764034&HistoricalAwards=false
• 关键词: Finite; time; blowup; supercritical; equations

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.

该项目涉及两个数学研究方向。第一个方向涉及对水波或声波等波的数学研究。物理实验和数值模拟都表现出湍流的现象：最初非常平滑且缓慢变化的流体可以发展出更为复杂的涡流和精细的行为。 尚不清楚模型流体动力学的方程是否会引起“爆炸”，其中一部分模拟的流体达到了无限速度。该项目通过调整模拟流体力学的方程式（例如更改空间中的尺寸数量）来研究爆炸可以“设计”的程度。第二个方向涉及与数字理论中臭名昭著的双素猜想有关的问题。这种猜想断言，有很多质子数的无限成对，它们的距离为两个距离，例如3和5、5、5和7、11和13、17和19，等等。 仍然不知道该断言是正确的，但是最近在理解更可行的问题方面已经取得了很大进展，例如发生了多久发生的一对数字，距离距离为两个距离，两者都有奇数的主要因素。该项目旨在进一步开发这些有希望的新技术，并潜在地证明或反驳了双胞胎素数的猜想，并在数字理论中提出了其他困难，重要的问题。更详细地，该项目的流体动力学部分主要集中在不可压缩的流体的Euler方程的变体上，尤其是Riemannian歧管上的较高尺寸的Euler方程。选择这种歧管的度量的能力为“编程”方程式表现出某些理想行为提供了一种有希望的方法。先前的工作已经确定，可以将某些二次普通微分方程的动力学编程到此类系统中。 该项目将继续致力于展示这些模型的有限时间爆炸（或其他有趣的行为，例如图灵普遍性）。对于项目的数量理论方面，研究人员和合作者旨在通过结合最近开发的熵递减方法与分析数理论，结合统计奖的nss thes ns ns ns ns ns ns ns ns ns ns ns ns thesfs n s thesfs n s n s n s n s n s thesf Reforces FordiS ns ns ns ns ns ns ns thtsf Reforces Forces Forces Forces FordiS theSf Repporce n s theSf Reppords Forces Forces Forces Forces Forces Forces Fordiss ns ns nsfore ns ns ns ns nsfors。通过基金会的智力优点和更广泛的影响评估标准通过评估来支持。