Mathematical Problems in Compressible Fluid Flow

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This project will address various mathematical questions relating to the existence, regularity, qualitative features, and large-time behavior of solutions of certain systems of partial differential equations. These systems are closely related mathematically, all sharing common features with the Navier-Stokes equations of multidimensional, compressible fluid flow and arise in a variety of applications, including the flow of compressible fluids and gases, shallow-water theory, magnetohydrodynamics, and the modeling of self-gravitating stellar systems. The primary goal of the project is to achieve precise, mathematically certain statements about these models by applying techniques from the fields of analysis and partial differential equations. This research seeks to impact the underlying physical science by validating the models, setting limits on the range of applicability of the models, or in some cases even invalidating the models under consideration. At the intellectual level it applies a reasoning process quite distinct from that in which laboratory data is compared with predictions based on numerical simulations. Indeed, the mathematical mechanisms which insure well-posedness of realistic models nearly always have physical counterparts; this means that the challenge to analyze these models with strict mathematical rigor is also a challenge to understand the physical phenomena at a newer, deeper level. Additionally, by identifying the correct framework in which a mathematical model is well-posed and the mathematical mechanisms which make it so, this research can point the way to the effective design of methods for generating approximate solutions by computer simulation. This project will address various mathematical questions relating to important physical models of compressible fluids and materials. These models arise in a broad range of applications, including supersonic flight, dynamic meteorology, combustion processes, and stellar dynamics. While the main goal in constructing these models is to achieve a predictive capability, the systems of equations which comprise them are far too complicated to be "solved" in any explicit sense. On the other hand, adequate approximate solutions can frequently be generated by computer methods. The intelligent and effective design of such methods depends crucially, however, on a rigorous understanding of why solutions do exist, in what sense they exist, and in what ways they are sensitive to noise in the data. The project will therefore seek to provide this rigorous mathematical framework for these models and to determine possible limits on the range of their applicability. A second goal of the project will be to attract and nurture the development of younger mathematicians to this area of research. Specifically, the PI will continue collaborative research with present and future graduate students and postdocs, will teach an advanced graduate-level course on mathematical problems in compressible flow in the second year of the grant period, and will continue supervising undergraduate research in Indiana University's REU program approximately every third summer.