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

这个项目涉及偏微分方程的数学研究。特别地，我们将研究平面上闭域上两种不同的线性偏微分算子的边值问题。第一个是跨声速流体流动的Tricomi方程，第二个是轴对称冷等离子体中电磁波传播的模型方程。这项工作的主要目的是通过研究这些问题的可能奇点的位置和强度，来描述在整个边界上规定数据的奇异解的适定问题。所采用的方法是现代微局部分析与经典对象（如特殊函数）的混合。将采用显式解算符来获得有关奇异点位置和强度的精确信息。这些模型所采用的显式方法将使我们对混合型问题有一个更全面的认识，并为混合型非线性边值问题的研究提供基础。此外，对这些模型中奇点的精确描述具有科学意义，因为它们的存在具有物理上合理的解释。在流体模型中，奇点对应于跨音速状态下可能存在的激波，而在等离子体模型中，奇点与电磁波传播中吸收能量可能引起的等离子体加热有关。偏微分方程是物理科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是说明关系，而是从中提取定性和定量的意义。

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.