Mathematical Sciences: Mathematical Problems in CompressibleFluid Flow

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The investigator will study various questions relating to the existence, stability, and numerical approximation of solutions of certain models of compressible fluid flow. Previous work concerning one-dimensional and spherically symmetric flows will be extended to general multidimensional flows and to models for chemically reacting flows and flows involving phase transitions. Two long-range goals of the project are: first, to prove the global existence of solutions of the Navier-Stokes equations for compressible flow with large, discontinuous Cauchy data; second, to understand solutions of these equations in the one-dimensional case, with Riemann initial data, in terms of solutions of the corresponding Euler equations for inviscid fluids. The mathematical models studied in this project attempt to predict the behavior of fluids which may react chemically so as to ignite, and fluids in which both the liquid or gaseous states may be present simultaneously. The design and development of computer methods for numerically solving the model equations is greatly facilitated by a rigorous mathematical understanding of why solutions do exist and in what sense, and how sensitive these solutions are to the initial state of the system. The primary goal of this project is to provide this rigorous mathematical understanding of the mathematical models; a secondary objective is to apply these mathematical results to the design, testing, and analysis of computer algorithms for generating approximate solutions.