Topics in fluid dynamics

流体动力学主题

• 批准号: 1311964
• 负责人: Walter Rusin
• 金额: $ 12.29万
• 依托单位: Oklahoma State 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: 2013-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1311964&HistoricalAwards=false
• 关键词: Topics; fluid; dynamics

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.

该项目涉及的主题涉及流体动力学研究中出现的偏微分方程。特别是，我们研究了不可压缩的Navier-Stokes方程的运动的粘性液体，欧拉方程（及其模型）的运动的无粘液体，一类标量方程自然产生的上述系统通过近似在物理上重要的设置。在第一部分中，关于Navier-Stokes方程，我们打算探讨的各向异性引入的振荡结构的解决方案，以构建类的大数据导致全球适定性。该项目的第二部分是面向获得更深入的了解一类活动标量方程的不适定性的基础上，第二个周期的性质。分析特别侧重于方程的非线性是由傅立叶乘数的符号是各向异性和/或甚至（多孔介质方程和磁地转方程）。本计画的第三个目标是研究一个偏微分方程系统，以模拟三维不可压缩欧拉方程中涡量拉伸现象的基本特徴。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。这些偏微分方程代表了许多自然现象和技术应用建模的基本工具。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，如最佳空气动力学和流体动力学形状的设计。流体动力学方程是出了名的难解，即使是在最大的计算机的帮助下。其中一些困难构成了问题的一部分，但另一些困难是由于我们对方程本身的数学理解不完整。从分析的观点来看，这些方程以高度非线性的方式解释了广泛的空间和时间尺度的相互作用。由于这种内在的复杂性，通过数值计算完全解决潜在现象所需的准确性在可预见的未来是遥不可及的。这些偏微分方程的理论理解的进展是至关重要的模型的验证和数值结果的准确解释。归根结底，理论研究的主要任务是找到控制解的行为的简单而自然的参数。一旦一组好的参数是已知的，它变得更容易设计实用的方法来计算数值的解决方案。