New trends in mathematical fluid dynamics and ergodic number theory

数学流体动力学和遍历数论的新趋势

• 批准号: 1265547
• 负责人: Yakov Sinai
• 金额: $ 20.4万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265547&HistoricalAwards=false
• 关键词: New; trends; mathematical; fluid; dynamics

Many phenomena in nature are connected with deep mathematical problems and theories. As an example, one can mention the obvious fact that in many regions on the Earth the frequency of tornadoes has strongly increased. Tornadoes are special solutions of equations of fluid dynamics that have strong singularities. A big part of the mathematical efforts detailed in this proposal is connected with the general properties of solutions of equations of fluid dynamics. Several years ago D. Li and I proposed a new approach which allows us to construct solutions which have singularities that in some respects resemble tornadoes. Recently a group of researchers from Italy performed numerical studies and demonstrated solutions that fully confirmed our theory. Another part of the proposal is connected with number theory. One of the main objects in number theory is the so-called Moebius function. P.Sarnak has proposed a wide program on Mobuius functions whose aim was the analysis of statistical properties of the Moebius functions. Some progress was achieved in our joint papers with F.Cellarosi and with M. Avdeeva and Dong Li. A new feature here is the appearance of probability distributions with complex probabilities which certainly is a new phenomenon. The purpose of the proposal is to study in more detail complex probabilities, new limit theorems and find other applications to number theory.The topics in this proposal center around questions in the fields of ergodic theory and the theory of partial differential equations. In broad terms, ergodic theory is the study of the asymptotic behavior of the paths of objects in motion. This field was borne through the study of statistical mechanics, and indeed ergodic theory remains a valuable tool in this important subject of physics. Another subject in which ergodic theory has found application is in number theory, and in particular in the study of the asymptotic distribution of certain kinds of numbers. The application of ergodic theory to number theory has itself led to fascinating questions in ergodic theory. This proposal contains a collection of such questions that we will continue to study. Partial differential equations are the principal tool by which physical phenomena, such as the flow of air over the wing of an airplane, the behavior of plasmas in the body of a star, the flow of electrons through a semi-conductor, are modeled. It is a subject which is intensively studied for its own inherent interest as well as for its usefulness in applications. We propose the continuation of deep studies in the properties of the equations of fluid dynamics as we described above.

自然界中的许多现象都与深奥的数学问题和理论有关。举个例子，人们可以提到一个明显的事实，即在地球上的许多地区，龙卷风的频率大大增加了。龙卷风是具有强奇异性的流体动力学方程的特殊解。在这个建议中详述的数学努力的很大一部分与流体动力学方程解的一般性质有关。几年前，李博士和我提出了一种新的方法，它允许我们构建具有奇点的解决方案，在某些方面类似于龙卷风。最近，来自意大利的一组研究人员进行了数值研究，并展示了完全证实我们理论的解决方案。提案的另一部分与数论有关。数论的主要对象之一是所谓的莫比乌斯函数。P.Sarnak提出了一个关于莫比乌斯函数的广义程序，其目的是分析莫比乌斯函数的统计性质。我们与塞拉罗西先生、阿夫季耶娃先生和李东的联合论文取得了一些进展。这里的一个新特征是出现了具有复杂概率的概率分布，这当然是一个新现象。该提案的目的是更详细地研究复杂概率，新的极限定理，并找到数论的其他应用。本提案的主题围绕遍历理论和偏微分方程理论领域的问题。广义地说，遍历理论是研究运动中物体路径的渐近行为。这个领域是通过对统计力学的研究而产生的，事实上，遍历理论在这一重要的物理学学科中仍然是一个有价值的工具。遍历理论的另一个应用领域是数论，特别是对某些数的渐近分布的研究。遍历理论在数论中的应用本身就导致了遍历理论中引人入胜的问题。这个建议包含了一系列我们将继续研究的问题。偏微分方程是物理现象建模的主要工具，如飞机机翼上的空气流动、恒星体内等离子体的行为、电子在半导体中的流动等。这是一门因其自身固有的兴趣以及在应用中的有用性而被深入研究的学科。如上所述，我们建议继续深入研究流体动力学方程的性质。