Regularity Problem on Two Models from Fluid Dynamics

WuDMS-1614246 研究人员研究了两个著名的偏微分方程模拟地球物理流体：表面准地转方程和磁流体动力学方程。 偏微分方程是理解许多流体现象的基本工具，在许多实际应用中起着举足轻重的作用。 地面准地转方程模拟大气和海洋的大尺度运动，如锋生，即热空气和冷空气之间尖锐锋的形成。 磁流体动力学方程模拟存在磁场的导电流体，例如等离子体、液态金属和电解质。 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。 研究这个问题的地面准地转方程导致更好地理解的形成和演变的剧烈的气象现象，如雷暴和龙卷风。 对磁流体动力学方程的研究揭示了磁重联和磁湍流的奇异行为。 磁场重联是指等离子体中反向磁场线的断裂和重新连接，是太阳系中许多壮观事件的核心，如太阳耀斑和北方极光。 研究生参与了该项目的工作。 这个项目解决了一些基本问题的表面准地转方程和磁流体动力学方程，包括这些方程的经典解决方案是否可以开发有限时间奇点的问题。 有效的技术和非常规的方法来理解这些模型的非线性和潜在的奇异和湍流动力学。 此外，进行了大量的数值计算来模拟各种解决方案的演变，提供洞察这些方程的解决方案的复杂行为。 该项目将研究与研究生和博士后学者的教育和培训相结合。

WuDMS-1614246 The investigator studies two well-known partial differential equations modeling geophysical fluids: the surface quasi-geostrophic equation and the magneto-hydrodynamic equation. Partial differential equations are fundamental tools in understanding many fluid phenomena, and they have played pivotal roles in many practical applications. The surface quasi-geostrophic equation models large-scale motions of atmosphere and oceans such as frontogenesis, the formation of sharp fronts between masses of hot and cold air. The magneto-hydrodynamic equation models electrically conducting fluids in the presence of a magnetic field, such as plasmas, liquid metals, and electrolytes. This project focuses on the fundamental problem of whether physically relevant solutions to these equations are globally regular for all time or they develop singularities. Studying this problem for the surface quasi-geostrophic equation leads to a better understanding of the formation and evolution of violent meteorological phenomena such as thunderstorms and tornadoes. Study of the magneto-hydrodynamic equation sheds light on the singular behavior of magnetic reconnection and magnetic turbulence. Magnetic reconnection refers to the breaking and reconnecting of oppositely directed magnetic field lines in a plasma and is at the heart of many spectacular events in our solar system, such as solar flares and northern lights. Graduate students are involved in the work of the project. This project addresses a number of fundamental problems concerning the surface quasi-geostrophic equation and the magneto-hydrodynamic equation, including the issue of whether classical solutions of these equations can develop finite-time singularities. Effective techniques and unconventional approaches are developed to understand the nonlinear and potentially singular and turbulent dynamics of these models. In addition, extensive numerical computations are carried out to simulate the evolution of various solutions, to provide insight into the perplexing behavior of solutions to these equations. The project integrates research with education and training of graduate students and postdoctoral scholars.