Analysis and Applications of Two Partial Differential Equations Modeling Geophysical Fluids

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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.