Two Partial Differential Equations Modeling Geophysical Fluids

Oklahoma State University, Stillwater OK

Investigators

Abstract

Wu DMS-0907913 This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The project focuses on two well-known partial differential equations modeling geophysical fluids: the surface quasi-geostrophic (SQG) equation and the two-dimensional Boussinesq equations. The major objective is to develop strategies and effective approaches for solving the global regularity problem on the classical solutions of these equations. The global regularity issue concerning these equations has recently attracted substantial attention and much important progress has been made. However, it remains open in the cases of the inviscid SQG equation, the SQG equation with supercritical dissipation, and the inviscid Boussinesq equations. To deal with the inviscid or supercritical SQG equation, the investigator combines extensive numerical computations with analytic and geometric approaches. The immediate plan is to study the curvature of the level curves in the spatial regions where the gradients are comparable to the maximal gradient. The boundedness of the curvature in these regions would rule out any finite-time singularities. The strategy on the global regularity issue for the two-dimensional Boussinesq equations is to gradually reduce the dissipation and thermal diffusion. The first aim is at the case when there is only vertical dissipation or thermal diffusion. In contrast to the recently resolved case with horizontal dissipation or thermal diffusion, the situation now is more sophisticated due to the "mismatch" of derivatives. To handle this case, new tools such as logarithmic type inequalities involving Sobolev norms of derivatives in different directions are developed. The three-dimensional quasi-geostrophic equations derived by J. G. Charney in the 1940s have been very successful in modeling large-scale motions of atmosphere and oceans. The dynamics of these three-dimensional equations with uniform potential vorticity reduces to the SQG equation. The SQG equation has been very useful in studying many weather phenomena such as frontogenesis, the formation of sharp fronts between hot and cold air. Mathematically, frontogenesis corresponds to the fundamental issue of whether classical solutions of this equation can develop finite-time singularities. This project helps improve the understanding of many weather phenomena governed by this equation. Boussinesq equations model many flows in nature such as oceanic circulation, central heating and natural ventilation. The study here of the potentially singular behavior of solutions to the Boussinesq equations not only yields a significant contribution to the mathematical issue of global regularity but also has potential environmental applications. As part of this project, several Ph.D. students of the investigator are actively involved in the proposed research and develop analytic and computational skills that enable them to become capable scholars and highly skilled workforce.