Mathematical Sciences: Nonlinear Partial Differential Equations

这个项目研究流体力学的几个数学方面。将对数值方法进行计算测试，称为涡流方法，用于随时间变化的、无粘的、不可压缩的流体流动。所使用的测试问题是带涡旋的涡环；其他研究人员可以使用变分方法可靠地计算它们。研究结果对涡旋法的应用具有一定的指导意义，并对涡旋法的改进提出了建议。将对流体界面相关方法中的错误进行分析，以期更好地了解它们的设计。在这种情况下，跟踪移动的界面，以及确定其速度的信息。我们将沿着涡旋法的早期分析的思路，尝试一个收敛证明。将继续研究由Korteweg-de Vries方程模拟的代表相互作用的长波的三维水波方程的精确解的构造工作。与通常尺度中的波的近乎共振导致了精细的分析。最后，将研究描述大尺度大气和海洋运动的方程与它们的奇异极限形式--准地转方程之间的关系。这种近似的性质与整个系统的解在时间或空间上产生不必要的小尺度密切相关。在流体流动的性质具有实际重要性的许多情况下，例如流经物体或燃烧的流动，由于涉及大量变量，因此很难进行定量预测。对于真实流动的数值模拟的可靠方法的发展提供了希望，可以用更少的努力来研究更多的测试用例，而不是仅仅通过实验。涡旋法是一类特殊的方法，在重要但困难的情况下具有独特的优势。当一种流体运动与另一种流体运动相遇时，就会出现界面。分析和数值模拟可以预测是否可以保持所需的配置。拟议的工作可能会对用于这些特殊问题的数值方法提出改进。在第三个主题中描述的这种类型的非线性波已被发现在水和其他物理环境中非常稳定；它们与线性理论中的波完全不同。人们对它们之间的完全非线性相互作用知之甚少。最后一个主题是对一个基本的和长期存在的问题的数学研究，即如何将大气或海洋中的大尺度环流方程简化为一个更简单的系统。人们不能指望准确地解决整个系统，人们已经投入了大量的努力，通过计算较少的变量来寻找预测主要特征的方法。

This project studies several mathematical aspects of fluid mechanics. Computational tests will be conducted for numerical methods, known as vortex methods, for time-dependent, inviscid, incompressible fluid flow. The test problems used are vortex rings with swirl; they can be computed reliably by other researchers using variational methods. The results may serve as a guide in the application of vortex methods and may suggest possible improvements in the methods. An analysis of the errors in related methods for fluid interfaces will be initiated, in the hope of better understanding their design. In such cases a moving interface is tracked, along with information determining its velocity. A convergence proof will be attempted along the lines of earlier analysis for vortex methods. Work on the construction of exact solutions for the equations of three-dimensional water waves, representing interacting long waves, of the type modeled by the Korteweg-de Vries equation, will be continued. Near-resonances with waves in the usual scaling have led to delicate analysis. Finally, the relationship between the equations describing large-scale atmospheric and oceanic motions and their singular limiting form, the quasi-geostrophic equations, will be investigated. The nature of this approximation is closely related to the generation of unwanted small scales in time or space in the solutions of the full system. In many situations where the nature of fluid flow is of practical importance, such as the flow past an object or combustion, quantitative prediction is difficult because of the large number of variables involved. The development of reliable methods for numerical simulation of realistic flows offers the hope that a greater number of test cases could be studied with less effort than through experiments alone. Vortex methods are a special class of methods which have particular advantages in important but difficult cases. An interface occurs where one kind of fluid motion meets another. Analysis and numerical simulation can predict whether or not desired configurations can be maintained. The proposed work might suggest improvements in the numerical methods used for these special problems. Nonlinear waves of the type described in the third topic have been found to be remarkably stable in water and in other physical settings; they are completely different from the waves occurring in linear theory. Little is known about their fully nonlinear interaction. The last topic is a mathematical investigation of the fundamental and long-standing question of how the equations for large-scale circulating flow in the atmosphere or oceans can be reduced to a simpler system. One cannot hope to solve the full system accurately, and much effort has been devoted to finding ways to predict the main features by calculating a smaller number of variables.