Mathematical Sciences: Singularities in Vortical Flows and Dispersive Systems

小行星9623087 这一建议是为了发展分析和计算方法 用于描述偏微分方程解的奇异性 （偏微分方程），并应用这些方法的问题，流体 机械和其他物理系统。分析的关键步骤是 - PDE的变量的变换以获得“展开”系统， 其解是非奇异的。这将用于涡面， 旋流，磁流体力学和晶体模式。 为这些 问题，我们将试图找到奇异的解决方案，并将其分类 泛型类型。 这一提议是为了研究物理学和工程学中的“奇点” 问题 奇点是宇宙中的可测量 系统突然改变或变得非常大。 一个重要的例子， 这是一个主要的重点，这个项目是发展非常大的， 在诸如空气或液体的流体中的旋转速度。 这些奇点是 被认为是湍流的主要原因，湍流是许多应用中重要但知之甚少的现象。 这项研究的目的是 项目是找到这些奇点，分类他们可能的行为， 用它们来了解湍流和其他问题。

9623087 Caflisch This proposal is for development of analytical and computational methods for describing singularities in the solutions of partial differential equations (PDEs), and for application of those methods to problems of fluid mechanics and other physical systems. The key step in the analysis is transformation of the variables of the PDE to obtain an "unfolded" system, the solutions of which are non-singular. This will be used for vortex sheets, swirling flow, magnetohydrodynamics and crystal patterns. For these problems we will attempt to find singular solutions and to classify their generic type. This proposal is for research on "singularities" in physics and engineering problems. Singularities are points at which the measurable quantities in the system change abruptly or become very large. An important example, which is a main focus of this project is the development of very large rotational velocity in a fluid such as air or liquid. These singularities are believed to be a primary cause of turbulence, an important but poorly understood phenomena in many applications. The goals of this research project are to find these singularities, classify their possible behavior and use them to gain understanding of turbulence and other problems.