Analysis and Applications of Two Partial Differential Equations Modeling Geophysical Fluids

该奖项将支持解决基本问题的研究，以及两个著名的偏微分方程建模地球物理流体的实际应用：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尺度运动，如锋生、冷空气和热空气之间锋面的形成等方面有着重要的应用。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方程也是有关湍流热对流的湍流理论的中心。在数学上，SQG方程和2D 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Navier-Stokes方程和三维Euler方程的关键特征，如涡拉伸机制。对这些2D模型的拟议研究可能会揭示围绕3D方程的奥秘。这个项目的重点是这些方程的物理相关的解决方案是否是全球定期的所有时间或他们开发奇点的基本问题。这里的目标是引入新的思想和发展有效的技术，以解决突出的开放的整体正则性问题的超临界SQG方程和Boussinesq方程的部分或无耗散。将进行广泛的大型数值模拟，以提供对潜在动力学的洞察力，并通过谐波分析和微分方程的专用工具进行新的估计，以衡量其解决方案的大时间行为。该奖项将支持博士学位。在俄克拉荷马州州立大学数学系的学生中，偏微分方程是理解从小尺度血液流动到大尺度地球物理流动的许多流体现象的基本工具，并且在涉及流体流动的许多实际应用中发挥了关键作用。其中最突出的是纳维尔-斯托克斯方程，它被广泛用于描述各种现象，如机器设备中的润滑、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。众所周知，研究和使用完整的Navier-Stokes方程是非常困难的，并且仍然存在与之相关的基本开放问题。因此，研究人员开发了这些方程的几个简化版本，仍然保留了它们的一些基本性质。这些偏微分方程已经在许多分析，实验和数值研究的中心。关于这些方程最突出的问题之一是，是否所有的经典解都是整体的时间。一个肯定的答案将表明，他们可能是有效的，甚至在极端的流动情况下，这是有意义的，试图解决他们的计算方法，而否定的答案将指出其使用的可能限制和固有的困难时，试图数值解。这个问题可能极具挑战性，在许多重要情况下仍然悬而未决。该项目的主要目标是创建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，并提高我们对这一难题的理解。重点将放在非常规的方法，结合联合收割机的方法和工具，从不同的数学学科，如流体力学，谐波分析和数值模拟。该项目将把研究与教育结合起来，促进教学和学生培训。四个博士学生将与PI合作解决该奖项将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将在颁奖期间组织几次专业科学会议和会议，这将涉及并支持初级数学家以及代表性不足的群体的研究人员。

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.