New trends in mathematical fluid dynamics and ergodic number theory

自然界中的许多现象都与深刻的数学问题和理论有关。 作为一个例子，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。龙卷风是具有强奇异性的流体动力学方程的特殊解。在这个建议中详细描述的数学努力的很大一部分与流体动力学方程的解的一般性质有关。几年前D。李和我提出了一种新的方法，使我们能够构造出在某些方面类似于龙卷风的奇点的解。最近，来自意大利的一组研究人员进行了数值研究，并证明了完全证实我们理论的解决方案。另一部分的建议是与数论。数论中的一个主要对象是所谓的Moebius函数。P.Sarnak提出了一个关于Moebius函数的广泛程序，其目的是分析Moebius函数的统计性质。我们与F.Cellarosi和M.阿芙迪娃和董丽。这里的一个新特征是具有复概率的概率分布的出现，这当然是一个新现象。该提案的目的是更详细地研究复杂的概率，新的极限定理，并找到数论的其他应用。该提案的主题围绕遍历理论和偏微分方程理论领域的问题。 广义上说，遍历理论是研究运动物体路径的渐近行为。 这一领域是通过统计力学的研究而产生的，事实上，遍历理论仍然是物理学这一重要学科中的一个有价值的工具。 遍历理论的另一个应用领域是数论，特别是研究某些数的渐近分布。 遍历理论在数论中的应用本身就引起了遍历理论中的一些有趣的问题。 这一建议包含了我们将继续研究的一系列此类问题。 偏微分方程是模拟物理现象的主要工具，例如飞机机翼上的空气流动、星星体内等离子体的行为、电子通过半导体的流动。 这是一个被深入研究的主题，因为它本身的内在利益以及它在应用中的有用性。 我们建议继续深入研究流体动力学方程的性质，正如我们上面所描述的。

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.