Two Partial Differential Equations Modeling Geophysical Fluids

WuDMS-0907913 该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。 该项目集中在两个著名的偏微分方程模拟地球物理流体：表面准地转（SQG）方程和二维Boussinesq方程。 本文的主要目的是为求解这类方程经典解的整体正则性问题提供策略和有效的方法。 关于这类方程的整体正则性问题近年来引起了人们的广泛关注，并取得了许多重要的进展. 然而，对于无粘SQG方程、具有超临界耗散的SQG方程和无粘Boussinesq方程，这一结论仍然是开放的。为了处理无粘或超临界SQG方程，研究者将大量的数值计算与解析和几何方法相结合。 目前的计划是研究梯度与最大梯度相当的空间区域中水平曲线的曲率。 这些区域中曲率的有界性将排除任何有限时间奇点。 解决二维Boussinesq方程整体正则性问题的策略是逐步减少耗散和热扩散。 第一个目标是在只有垂直耗散或热扩散的情况下。 与最近解决的水平耗散或热扩散的情况相反，由于导数的“不匹配”，现在的情况更加复杂。为了处理这种情况，新的工具，如对数型不等式涉及Sobolev范数的衍生物在不同的方向发展。 本文将J. G. Charney在20世纪40年代已经非常成功地模拟了大气和海洋的大尺度运动。 这些具有均匀位涡的三维方程的动力学性质归结为SQG方程。 SQG方程在研究锋生、冷空气与热空气之间锋面的形成等天气现象中有着重要的作用。 在数学上，锋生对应于这个方程的经典解是否可以发展有限时间奇点的基本问题。 这个项目有助于提高对这个方程所支配的许多天气现象的理解。 Boussinesq方程模拟了自然界中的许多流动，如海洋环流、集中供暖和自然通风。 对Boussinesq方程解的潜在奇异性的研究不仅对数学问题的全局正则性做出了重大贡献，而且具有潜在的环境应用价值. 作为该项目的一部分，几个博士。该大学的学生积极参与拟议的研究和发展分析和计算技能，使他们能够成为有能力的学者和高技能的劳动力。

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.