Mathematical Sciences: Boundary Value Problems for Operatorsof Mixed Type on Closed Domains

这个项目涉及部分的数学研究 微分方程 特别是，将开展以下工作： 两个不同线性偏微分方程的边值问题 平面闭域上的微分算子。 第一 是跨音速流体流动的Tricomi方程， 是电磁波传播的模型方程， 轴对称冷等离子体 这项工作的主要目标是 描述奇异解的适定性问题， 通过研究位置在整个边界上规定的数据 和可能的奇异性的强度。的 所采用的方法是现代微观局部的混合物， 分析经典对象，如特殊功能。 将采用显式解算子来获得精确解。 关于所产生的爆炸地点和强度的资料 奇点 这些模型中使用的显式方法应该 导致对混合类型问题的更一般的理解， 为非线性边值问题的研究奠定了基础 混合型问题。 此外，精确描述 这些模型中的奇点具有科学意义 因为它们的存在有着物理上合理的解释。 在流体模型中，奇点对应于可能的 在跨音速区域和等离子体中激波的存在 模型，奇点与可能的等离子体加热有关， 从电磁波传播中吸收能量。 偏微分方程是 物理科学中的数学建模。 的现象 包括连续变化，如运动、材料 和能量都服从某些普遍规律， 可以用相互作用和关系来表达 偏导数之间的关系 数学的关键作用不是 来陈述这些关系，而是为了提取定性的 和量化的意义。

This project concerns mathematical research on partial differential equations. In particular, work will be done on boundary value problems for two different linear partial differential operators on closed domains in the plane. The first is the Tricomi equation for transonic fluid flow and the second is a model equation for electromagnetic wave propagation in axisymmetric cold plasmas. The primary objective of the work is to characterize well-posed problems for singular solutions with data prescribed on the entire boundary by studying the location and strength of possible singularities to these problems. The methods to be employed are a mixture of modern microlocal analysis with classic objects such as special functions. Explicit solution operators will be employed to obtain precise information on the location and strength of the resulting singularities. The explicit methods used in these models should result in a more general understanding of mixed type problems and provide a foundation for the study of nonlinear boundary value problems of mixed type. In addition, the precise description of the singularities in these models is of scientific interest because their presence has physically reasonable interpretations. In the fluid models, singularities correspond to the possible presence of shock waves in transonic regimes, and in the plasma models, singularities are related to possible plasma heating due to energy absorption from the electromagnetic wave propagation. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.