FRG: Collaborative Research: Multi-Dimensional Problems for the Euler Equations of Compressible Fluid Flow and Related Problems in Hyperbolic Conservation Laws

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ABSTRACTFRG: Multi-Dimensional Problems for the Euler Equations of Compressible Fluid Flow and Related Problems in Hyperbolic Conservation LawsHistorically, fluid and solid mechanics study the motion ofincompressible and compressible materials, with or without internaldissipation. For gases and solids with internal dissipation as asecondary effect, the gross wave dynamics is governed by inviscid,thermal diffusionless, dynamics. Within these categories, compressiblemotion for solids corresponds to the study of elastic waves and theirpropagation; compressible motion for fluids is usually associated withinviscid gas dynamics. Furthermore both compressible solids and fluids exhibit shock waves and hence we must search for discontinous solutions to the underlying equations of motion.Incompressible motion on the other hand concernsitself with the motion of denser fluids where the idealization ofincompressibility is useful, e.g. water or oil, as well as the motion ofcertain solids like rubber. While there are still many importantmathematical issues to be resolved for incompressible fluids, for example,the well-posedness of the Navier-Stokes equations in three spacedimensions, the mathematical study of compressiblesolids (as represented by the equations of nonlinear elastodynamics) andfluids (as represented by the Euler equations of inviscid flows)in two and three space dimensions is even less developed.This provides the motivation to the proposers to collaborate in athree year effort to advance the mathematical understanding of themulti-dimensional equations of inviscid compressible fluid dynamicsand related problems in elastodynamics.The core of our plan is to arrange a sustained interaction between andaround the members of the group, who will(1) collaborate scientifically, focusing on the advancement of theanalysis of multi-dimensional compressible flows by developing newtheoretical techniques and by using and designing effective, robust andreliable numerical methods;(2) work together over the next several years to create the environmentand manpower necessary for the research on multi-dimensional compressibleEuler equations and related problems to flourish; and in the meantime,(3) share the responsibility of training graduate students andpostdoctoral fellows.The project is devoted to a mathematical study of the Euler equationsgoverning the motion of an inviscid compressible fluid and relatedproblems. Compressible fluids occur all around us in nature, e.g. gasesand plasmas, whose study is crucial to understanding aerodyanmics,atmospheric sciences, thermodynamics, etc.While the one-dimensional fluid flows are rather well understood, thegeneral theory for multi-dimensional flows is comparatively mathematicallyunderdeveloped. The proposers will collaborate in a threeyear effort to advance the mathematical understanding of themulti-dimensional equations of inviscid compressible fluid dynamics.Success in this project will advance knowledge of this fundamental area ofmathematics and mechanics and will introduce a new generation ofresearchers to the outstanding problems in the field.