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

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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.