Regularity Problem on Two Models from Fluid Dynamics

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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.