Mathematical Sciences: Nonlinear Partial Differential Equations

这个项目研究流体的几个数学方面 力学 计算测试将进行数值 方法，称为涡方法，用于时间依赖的，无粘性的， 不可压缩流体流动 使用的测试问题是涡流 环与漩涡;他们可以可靠地计算其他 研究人员使用变分方法。 结果可以作为 涡流方法的应用指南，并可能建议 方法上的可能改进。 错误分析 在流体界面的相关方法中， 希望能更好地理解他们的设计。 在这种情况下， 跟踪运动界面，沿着确定信息 它的速度。 将尝试沿着 涡方法的早期分析路线。 工作 方程的精确解的构造 三维水波，代表相互作用 长波浪，由Korteweg-de弗里斯模拟的类型 方程，将继续。 与波的近共振 通常的缩放导致了精细的分析。 最后 描述大尺度的方程之间的关系 大气和海洋运动及其奇异极限形式， 准地转方程。 的 这种近似的性质是密切相关的生成 在时间或空间上的不必要的小尺度， 整个系统。 在许多情况下，流体流动的性质 具有实际重要性，例如流过物体或 燃烧，定量预测是困难的，因为 涉及到的变量太多了。 的发展 真实流动数值模拟的可靠方法 提供了更多测试案例的希望， 比单独通过实验更容易研究。 涡方法是一类特殊的方法， 在重要但困难的案件中具有特殊优势。 一种流体运动相遇的地方， 另 分析和数值模拟可以预测 无论是否可以保持期望的配置。 拟议的工作可能会建议改进数字 解决这些特殊问题的方法。 非线性波 在第三个主题中描述的类型已经被发现 在水中和其他物理环境中非常稳定， 设置;他们是完全不同的波 发生在线性理论中。 知之甚少 完全非线性的相互作用。 最后一个主题是 一个数学调查的基本和 一个长期存在的问题， 大气或海洋中的循环流动 简化为一个简单的系统。 我们不能指望 整个系统的准确性，并已付出很大努力， 致力于寻找方法来预测 通过计算更少的变量。

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.