Analysis and Applications of Two Partial Differential Equations Modeling Geophysical Fluids

该奖项将支持研究解决基本问题以及模拟地球物理流体的两个著名的偏微分方程的实际应用：表面准地转（SQG）方程和二维（2D） Boussinesq方程。SQG方程在模拟大气和海洋的大规模运动方面非常有用，例如锋生，即冷热空气团之间尖锐锋的形成。Boussinesq方程也成功地模拟了许多地球物理流动，如大气锋和海洋环流。此外，Boussinesq方程也是湍流热对流理论的核心。在数学上，SQG方程和二维Boussinesq方程作为三维（3D）流体动力学方程的低维模型。事实上，它们都保留了三维纳维-斯托克斯方程和三维欧拉方程的关键特征，如涡旋拉伸机制。对这些二维模型提出的研究可能会揭示围绕三维方程的神秘面纱。该项目关注的基本问题是，这些方程的物理相关解是否始终是全局正则的，或者它们是否发展为奇点。本文的目的是引入新的思想和发展有效的技术来解决超临界SQG方程和部分或无耗散的Boussinesq方程上突出的开全局正则性问题。将进行广泛的大型数值模拟，通过谐波分析的专用工具提供对潜在动力学和新估计的见解，并推导微分方程以衡量其解的大时间行为。该奖项将支持俄克拉荷马州立大学的数学博士生。偏微分方程是理解从小规模血液流动到大规模地球物理流动等许多流体现象的基本工具，在涉及流体流动的许多实际应用中起着关键作用。其中最突出的是Navier-Stokes方程，它被广泛用于描述各种各样的现象，如机器设备的润滑、风力涡轮机周围的气流、海洋流动以及造成冷锋和急流的大规模大气流动。研究和使用完整的Navier-Stokes方程是出了名的困难，与之相关的基本问题仍然存在。因此，研究人员开发了这些方程的几个简化版本，这些方程仍然保留了一些基本性质。这些偏微分方程一直是许多分析、实验和数值研究的中心。这些方程最突出的问题之一是它们的经典解在时间上是否都是全局的。一个肯定的答案将表明，即使在极端的流动情况下，它们也可能是有效的，并且尝试用计算方法来解决它们是有意义的，而一个否定的答案将指出它们的使用可能存在局限性，并且在尝试数值解决时存在固有的困难。这个问题极具挑战性，在许多重要的案例中仍未解决。该项目的主要目标是创造新的策略和开发有效的技术，以进一步推进对这一难题的研究和提高我们对这一难题的理解。重点将放在结合不同数学学科如流体力学、谐波分析和数值模拟的方法和工具的非常规方法上。该项目将把研究与教育结合起来，促进教学和培养学生。四名博士生将与PI一起研究该奖项所要解决的问题的各个方面，并将接受使用各种数学技术的培训。此外，PI将在奖励期间组织几次专门的科学会议和会议会议，这些会议将涉及和支持初级数学家以及来自代表性不足群体的研究人员。

This award will support research that addresses fundamental issues as well as practical applications of two well-known partial differential equations modeling geophysical fluids: the surface quasi-geostrophic (SQG) equation and the two-dimensional (2D) Boussinesq equation. The SQG equation has been very useful in modeling large-scale motions of atmosphere and oceans such as the frontogenesis, the formation of sharp fronts between masses of hot and cold air. The Boussinesq equation has also been successful in modeling many geophysical flows such as atmospheric fronts and ocean circulations. In addition, the Boussinesq equation is also at the center of turbulence theories concerning turbulent thermal convection. Mathematically, the SQG equation and the 2D Boussinesq equation serve as lower dimensional models of the three-dimensional (3D) hydrodynamics equations. In fact, they both retain the key features of the 3D Navier-Stokes and the 3D Euler equations such as the vortex stretching mechanism. The proposed study on these 2D models may shed light on the mystery surrounding the 3D equations. This project focuses on the fundamental issue of whether physically relevant solutions to these equations are globally regular for all time or they develop singularities. The objective here is to introduce new ideas and develop effective techniques to solve the outstanding open global regularity problem on the supercritical SQG equation and on the Boussinesq equation with partial or no dissipation. Extensive large numerical simulations will be performed to provide insight to the underlying dynamics and novel estimates via dedicated tools from harmonic analysis and differential equations will be derived to gauge the large-time behavior of their solutions. The award will support Ph.D. students in mathematics at Oklahoma State University.Partial differential equations are fundamental tools in understanding many fluid phenomena ranging from small scale blood flows to large-scale geophysical flows and have played pivotal roles in many practical applications involving fluid flows. Outstandingly among them are the Navier-Stokes equations, which are widely used for describing phenomena that are as varied as lubrication in machine equipment, airflow around a wind turbine, oceanic flows, and large scale atmospheric flows that are responsible for cold fronts and the jet stream. The study and use of the full Navier-Stokes equation is notoriously difficult and there are still fundamental open questions associated with them. Researchers have therefore developed several simplified versions of these equations that still retain some of their fundamental properties. These partial differential equations have been at the center of numerous analytical, experimental and numerical investigations. One of the most prominent problems concerning these equations is whether all of their classical solutions are global in time. An affirmative answer would suggest that they may be valid even in extremal flow situations and that it makes sense to attempt to solve them with computational methods, while a negative answer will point to possible limitations of their use and to inherent difficulties when a numerical solution is attempted. This problem can be extremely challenging and remains open in many important cases. The main goal of this project is to create new strategies and develop effective techniques to further advance the research on and improve our understanding of this difficult problem. The emphasis will be on unconventional approaches that combine methods and tools from different mathematical disciplines such as fluid mechanics, harmonic analysis and numerical simulations. The project will integrate research with education, promoting teaching and training of students. Four Ph.D. students will work with the PI on various aspects of the problems that will be tackled with this award and will be trained in the use of a wide variety of mathematical techniques. In addition, the PI will organize several specialized scientific conferences and conference sessions during the award period, which will involve and support junior mathematicians as well as researchers from under-represented groups.