Applied Analysis of the Navier-Stokes and Related Equations

这是一个数学物理和应用数学基础研究的建议，主要针对流体力学中不可压缩的Navier-Stokes方程和相关方程所带来的挑战。Navier-Stokes方程构成了窗口的基本数学模型，并且被认为在其解中包含了湍流动力学。流体力学中的湍流仍然是理论物理和应用数学面临的突出挑战之一，在物理科学和工程的许多领域都有重要的应用和影响。该项目的工作将通过现代应用分析和数值计算与模拟进行，由首席研究员（PI），密歇根大学的Charles R. doering教授和研究生进行博士论文工作。该项目有三个具体目标：首先，一种推导湍流耗散和阻力严格界限的数学技术，即所谓的“背景法”，将被发展并扩展到新的领域，包括磁流体动力学、减阻聚合物模型和粗糙边界上的模型。背景方法是由PI和他的合作者在十年前为Navier-Stokes方程引入的，从那时起，PI、他的学生和许多其他研究人员发展并应用于许多基本的剪切流和热对流问题。这个项目的一个特别目标将是探索应用于更广泛的科学兴趣问题。本项目的另一个重点是继续研究热对流模型分析中的理论和数学问题，其中背景方法能够限制传热速率。这里关注的问题包括具有自由滑移边界的层流和湍流对流，x -ux对流，由内部加热驱动的流，以及受地球物理学应用启发的无限普朗特数模型。在研究的第三个方向上，将研究在没有刚性边界的情况下，强制流动和自由流动的电力消耗和能量产生。PI和合作者最近开发了一种新的方法来分析由时间无关的体力驱动的湍流，并建议将结果扩展到时间相关的力。非强迫涡的一个独特问题是解决最大熵生成构型的变分问题，并研究它与在完全发展的湍流或奇点的潜在发展中观察到的结构的关系。关于这项活动的智力价值，从这个项目中获得的知识将进一步加深我们对流体动力学中与许多工程和应用科学分支直接相关的一些基本数学模型的理解。从长远来看，这种数学研究可以帮助开发实用技术，用于预测和/或控制从气象学到材料制造等物理过程。关于这项活动的广泛影响，该项目有几个重要的高级培训方面。首先，它为密歇根大学应用与跨学科数学新博士项目的研究生提供研究支持和机会。此外，该项目还涉及其他研究人员，包括密歇根大学和其他机构的研究生和博士后研究人员，他们将在研究中进行合作。

This is a proposal for fundamental research in mathematical physics and applied mathematics fo-cused on the challenges presented by the incompressible Navier-Stokes and related equations ofuid mechanics. The Navier-Stokes equations constitute the basic mathematical model of uidow, and are believed to contain turbulent dynamics among their solutions. Turbulence in uidmechanics remains one of the outstanding challenges for theoretical physics and applied mathe-matics with important applications in, and implications for, many areas in the physical sciencesand engineering. The work in this project will be carried out via modern applied analysis andnumerical computation and simulation by the principal investigator (PI), Prof. Charles R. Doeringof the University of Michigan, and graduate students performing doctoral dissertation work. Thisproject has three specific objectives:For one, a mathematical technique for deriving rigorous bounds on turbulent dissipation anddrag, which has come to be known as the \background method," will be developed and expandedto new areas including magnetohydrodynamics, drag-reducing polymer ows, and ows over roughboundaries. The background method was introduced by the PI and his collaborator a decade agofor the Navier-Stokes equations, and since then it has been developed and applied by the PI, hisstudents, and many other researchers to a number of fundamental shear ow and thermal convectionproblems. One particular goal of this project will be to explore applications to a wider variety ofproblems of scientific interest.Another focus of this project is to continue the ongoing investigation of theoretical and mathe-matical issues in the analysis of thermal convection models where the background method is capableof putting limits on the heat transfer rate. Problems of concern here include laminar and turbulentconvection with free-slip boundaries, xed-ux convection, ows driven by internal heating, andinfinite Prandtl number models inspired by applications in geophysics.In a third direction of research, power consumption and enstrophy generation will be studiedfor forced ows and free ows in the absence of rigid boundaries. The PI and collaborators haverecently developed a new approach for the analysis of turbulence driven by time-independent body-forces, and it is proposed to extend the results to time-dependent forces. A distinct problem forunforced ows is to solve a variational problem for the maximum enstrophy-generating configurationand study how it relates to structures observed in fully developed turbulence or the potentialdevelopment of singularities.With regard to the intellectual merit of this activity, knowledge gained from this project willfurther our understanding of some basic mathematical models in uid dynamics of direct relevanceto many branches of engineering and applied science. In the long term, this kind of mathematicalresearch could help the development of practical techniques for the prediction and/or control ofphysical processes ranging from meteorology to materials manufacturing.And with regard to this activity's broader impacts, there are several significant advanced train-ing aspects to the project. For one, it provides research support and opportunities for graduatestudents within the University of Michigan's new Ph.D. program in Applied & InterdisciplinaryMathematics. Moreover, this project also involves other investigators|including graduate studentsand postdoctoral researchers from the University of Michigan as well as other institutions|whowill collaborate in the research.