Mathematical Problems in Compressible Fluid Flow

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This project investigates questions relating to the existence, stability, regularity, and qualitative features of certain systems of partial differential equations. These systems are closely related mathematically, all sharing common features with the Navier-Stokes equations of compressible fluid flow, and all serving as models for physical phenomena arising in concrete applications, including the flow of compressible fluids and gases, shallow water theory, magnetohydrodynamics, combustion theory, and semiconductor modeling. The primary goal is to develop precise, mathematically certain statements about these models by applying techniques from the fields of analysis and partial differential equations. This research is related to the underlying physical science in several ways: first, it can validate, determine limits on the range of applicability, or in some cases invalidate the models; second, rigorous mathematical analysis of the models can lead to deeper understanding of the physical phenomena; finally, by determining topologies in which the models are well-posed and elucidating underlying reasons for this structure, the work can set the stage for the development of computer procedures for the effective computation of approximate solutions.The proposer will study mathematical questions concerning important models of compressible fluids and materials. These models arise in a broad range of applications, including supersonic flight, dynamic meteorology, semiconductor theory, and combustion processes. While the main goal in constructing these models is to achieve a predictive capability, they are far too complicated to be solved in any explicit sense. On the other hand, adequate approximate solutions can frequently be generated by computational methods. However, the intelligent design of such methods depends crucially on understanding of why solutions do exist, in what sense, and in what ways they are sensitive to noise in the data. The project will provide such a rigorous mathematical analysis for these models and determine possible limits on their range of applicability, thereby setting the stage for the development of computer procedures for the effective computation of approximate solutions.