Topics in fluid dynamics

本计划所讨论的主题涉及流体动力学研究中出现的偏微分方程。特别地，我们研究了粘性液体运动的不可压缩Navier-Stokes方程，非粘性液体运动的欧拉方程（及其模型），以及通过在物理重要设置中的近似自然产生的一类标量方程。在第一部分中，关于Navier-Stokes方程，我们打算探讨由解的振荡结构引入的各向异性，以便构造导致全局适定性的大数据类。该项目的第二部分是基于第二次迭代的性质，对一类活动标量方程的病态性有更深的理解。分析的重点是非线性由符号为各向异性和/或偶数的傅里叶乘数给出的方程（多孔介质方程和磁地转方程）。本项目的第三个目标是研究一个模拟三维不可压缩欧拉方程中涡度拉伸现象基本特征的偏微分方程系统。该项目旨在通过严格的数学分析方法，促进对流体动力学方程所描述的动力学的理解。这些偏微分方程是模拟许多自然现象和技术应用的基本工具。例如，描述流体流动的方程（这是本项目的一个重要部分）用于天气预报、气候研究和各种技术应用，例如设计最佳的空气动力和水动力形状。众所周知，流体动力学方程很难解，即使有最大的计算机的帮助。其中一些困难构成了问题的一部分，但有些困难是由于我们对方程本身的数学理解不完整。从分析的角度来看，这些方程以高度非线性的方式解释了大范围的空间和时间尺度的相互作用。由于这种内在的复杂性，在可预见的未来，通过数值计算完全解决潜在现象所需的精度是遥不可及的。对这些偏微分方程的理论理解的进步对于模型的验证和数值结果的准确解释是至关重要的。归根结底，理论研究的主要任务是找到控制解行为的简单而自然的参数。一旦知道了一组很好的参数，设计实用的数值计算方法就容易多了。

The topics addressed in the project concern partial differential equations arising in the study of fluid dynamics. In particular, we study the incompressible Navier-Stokes equations for the motion of a viscous liquid, the Euler equations (and their models) for the motion of an inviscid liquid, and a class of scalar equations arising naturally from the mentioned systems through approximations in physically important settings. In the first part, regarding the Navier-Stokes equations, we intend to explore the anisotropy introduced by the oscillatory structure of solutions, in order to construct classes of large data leading to global well-posedness. The second part of the project is geared towards gaining a deeper understanding of the ill-posedness of a class of active scalar equations, based on the properties of the second iterate. The analysis focuses particularly on the equations where the nonlinearity is given by a Fourier multiplier whose symbol is anisotropic and/or even (the porous media equation and the magneto-geostrophic equation). The third objective of this project is to investigate a system of PDEs simulating the fundamental features of the vorticity-stretching phenomena in the three-dimensional incompressible Euler equations. The project seeks to advance the understanding of the dynamics described by the equations of fluid dynamics by methods of rigorous mathematical analysis. These partial differential equations represent a basic tool for modeling many natural phenomena and technological applications. For instance, the equations describing fluid flow (on which an important part of this project concentrates) are used in weather prediction, climate research, and various technological applications, such as the design of optimal aerodynamical and hydrodynamical shapes. The equations of fluid dynamics are notoriously difficult to solve, even with the help of the largest computers. Some of these difficulties constitute part of the problem, but some are due to the fact that our mathematical understanding of the equations themselves is incomplete. From the analytical point of view, these equations account for interactions of a broad range of space- and time-scales in a highly non-linear fashion. Due to this intrinsic complexity, the accuracy needed to fully resolve the underlying phenomena via numerical computations is out of reach for the foreseeable future. Advances in theoretical understanding of these partial differential equations are vital for the validation of the models and for an accurate interpretation of numerical results. In the final analysis, the main task of the theoretical investigation is to find simple and natural parameters that control the behavior of the solutions. Once a good set of such parameters is known, it becomes much easier to design practical methods for calculating the solutions numerically.