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

研究人员开发和应用有效的数学分析和科学计算工具，系统地搜索物理流体力学的一些基本方程中的极端行为。 目标是从第一原理中推导出物理上重要的量的精确预测。 这项工作利用了最近的发展，实现最优控制理论和变分法的思想，以计算流体流量，实现最大混合，最佳运输，或其他极端耗散。 传输、混合和耗散是流体流动的最基本特征之一，对于从微流体工程到气候科学和天体物理学建模的重要应用具有基础性意义。 这里采用的控制和优化技术构成了一个新的和统一的计算辅助分析方法，这些问题。 该项目直接涉及研究生和博士后研究人员的高级培训。该项目利用现代应用数学和科学计算方法。 研究者和合作者介绍的混合的数学措施被用于对流和对流扩散方程的最优控制分析，以绝对限制被动示踪剂混合不可压缩流，并照亮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来构造不可压缩流体流动，优化不可穿透表面之间的传输，并产生浮力驱动的Rayleigh-Benard对流和湍流对流的突出问题的新的传输边界。 最优控制技术的开发和部署，以确定最大涡度拟能生产的不可压缩三维Navier-Stokes方程在有限的时间间隔。 极值解为非强迫流中完全非线性涡量放大提供了新的见解，该项目的这一组成部分是研究21世纪世纪应用数学的信号挑战之一的新颖且有前途的框架：3D 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.