Two Partial Differential Equations Modeling Geophysical Fluids

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WuDMS-0907913 This award is funded under the American Recovery andReinvestment Act of 2009 (Public Law 111-5). The project focuseson two well-known partial differential equations modelinggeophysical fluids: the surface quasi-geostrophic (SQG) equationand the two-dimensional Boussinesq equations. The majorobjective is to develop strategies and effective approaches forsolving the global regularity problem on the classical solutionsof these equations. The global regularity issue concerning theseequations has recently attracted substantial attention and muchimportant progress has been made. However, it remains open inthe cases of the inviscid SQG equation, the SQG equation withsupercritical dissipation, and the inviscid Boussinesq equations. To deal with the inviscid or supercritical SQG equation, theinvestigator combines extensive numerical computations withanalytic and geometric approaches. The immediate plan is tostudy the curvature of the level curves in the spatial regionswhere the gradients are comparable to the maximal gradient. Theboundedness of the curvature in these regions would rule out anyfinite-time singularities. The strategy on the global regularityissue for the two-dimensional Boussinesq equations is togradually reduce the dissipation and thermal diffusion. Thefirst aim is at the case when there is only vertical dissipationor thermal diffusion. In contrast to the recently resolved casewith horizontal dissipation or thermal diffusion, the situationnow is more sophisticated due to the "mismatch" of derivatives. To handle this case, new tools such as logarithmic typeinequalities involving Sobolev norms of derivatives in differentdirections are developed. The three-dimensional quasi-geostrophic equations derived byJ. G. Charney in the 1940s have been very successful in modelinglarge-scale motions of atmosphere and oceans. The dynamics ofthese three-dimensional equations with uniform potentialvorticity reduces to the SQG equation. The SQG equation has beenvery useful in studying many weather phenomena such asfrontogenesis, the formation of sharp fronts between hot and coldair. Mathematically, frontogenesis corresponds to thefundamental issue of whether classical solutions of this equationcan develop finite-time singularities. This project helpsimprove the understanding of many weather phenomena governed bythis equation. Boussinesq equations model many flows in naturesuch as oceanic circulation, central heating and naturalventilation. The study here of the potentially singular behaviorof solutions to the Boussinesq equations not only yields asignificant contribution to the mathematical issue of globalregularity but also has potential environmental applications. Aspart of this project, several Ph.D. students of the investigatorare actively involved in the proposed research and developanalytic and computational skills that enable them to becomecapable scholars and highly skilled workforce.