Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics

研究者开发并应用有效的数学分析和科学计算工具，系统地搜索一些物理流体力学基本方程中的极端行为。目标是从第一性原理推导出物理量的精确预测。这项工作利用了最近的发展来实现最优控制理论和变分法的思想，以计算达到最大混合、最佳输送或其他极端耗散的流体流动。输运、混合和耗散是流体流动的最基本特征，对于从微流体工程到气候科学和天体物理学的建模等重要应用具有基础意义。本文采用的控制和优化技术为这些问题提供了一种新的、统一的计算辅助分析方法。本项目直接涉及研究生和博士后研究人员的高级培训。本项目运用现代应用数学和科学计算方法。研究者和合作者引入的混合数学方法被用于平流和平流扩散方程的最优控制分析，以便对不可压缩流动的被动示踪剂混合施加绝对限制，并阐明特别有效的搅拌策略的关键特征。采用计算控制和应用分析方法构建了不可压缩流体流动，优化了不可穿透表面之间的输运，并为浮力驱动的瑞利-贝纳德对流和湍流对流的突出问题建立了新的输运界。开发并应用了最优控制技术来确定有限时间间隔内不可压缩三维Navier-Stokes方程的最大熵产。极值解提供了对非强制流动中完全非线性涡度放大的新见解，该项目的这一组成部分是研究21世纪应用数学信号挑战之一的一个新颖而有前途的框架：三维Navier-Stokes方程的正则性问题。

The investigator develops and applies effective mathematical analysis and scientific computation tools to systematically search for extreme behavior in some of the fundamental equations of physical fluid mechanics. The goal is to derive precise predictions of physically significant quantities from first principles. The work capitalizes on recent developments to implement ideas from optimal control theory and the calculus of variations to compute fluid flows achieving maximal mixing, optimal transport, or other extreme dissipation. Transport, mixing, and dissipation are among the most fundamental features of fluid flows and are of foundational significance for important applications ranging from microfluidics engineering to modeling in climate science and astrophysics. The control and optimization techniques adopted here constitute a new and unified computationally aided analysis approach to these problems. This project directly involves advanced training for graduate students and postdoctoral researchers.This project utilizes methods of modern applied mathematics and scientific computation. Mathematical measures of mixing introduced by the investigator and collaborators are utilized in optimal control analyses of the advection and advection-diffusion equations in order to place absolute limits on passive tracer mixing by incompressible flows, and to illuminate key features of particularly effective stirring strategies. Computational control and applied analysis are employed to construct incompressible fluid flows optimizing transport between impenetrable surfaces and produce new transport bounds for buoyancy-driven Rayleigh-Benard convection and the outstanding problem of turbulent convection. Optimal control techniques are developed and deployed to determine maximal enstrophy production in the incompressible three-dimensional Navier-Stokes equations over finite time intervals. Extremal solutions provide new insight into fully nonlinear vorticity amplification in unforced flows, and this component of the project is a novel and promising framework for the study of one of the signal challenges for 21st century applied mathematics: the regularity question for the 3D Navier-Stokes equations.