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

ABSTRACT FRG: Multi-Dimensional Problems for the Euler Equations of Compressible Fluid Flow and Related Problems in Hyperbolic Conservation Laws Historically, fluid and solid mechanics study the motion of incompressible and compressible materials, with or without internal dissipation. For gases and solids with internal dissipation as a secondary effect, the gross wave dynamics is governed by inviscid, thermal diffusionless, dynamics. Within these categories, compressible motion for solids corresponds to the study of elastic waves and their propagation; compressible motion for fluids is usually associated with inviscid 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 concerns itself with the motion of denser fluids where the idealization of incompressibility is useful, e.g. water or oil, as well as the motion of certain solids like rubber. While there are still many important mathematical issues to be resolved for incompressible fluids, for example, the well-posedness of the Navier-Stokes equations in three space dimensions, the mathematical study of compressible solids (as represented by the equations of nonlinear elastodynamics) and fluids (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 a three year effort to advance the mathematical understanding of the multi-dimensional equations of inviscid compressible fluid dynamics and related problems in elastodynamics. The core of our plan is to arrange a sustained interaction between and around the members of the group, who will (1) collaborate scientifically, focusing on the advancement of the analysis of multi-dimensional compressible flows by developing new theoretical techniques and by using and designing effective, robust and reliable numerical methods; (2) work together over the next several years to create the environment and manpower necessary for the research on multi-dimensional compressible Euler equations and related problems to flourish; and in the meantime, (3) share the responsibility of training graduate students and postdoctoral fellows. The project is devoted to a mathematical study of the Euler equations governing the motion of an inviscid compressible fluid and related problems. Compressible fluids occur all around us in nature, e.g. gases and plasmas, whose study is crucial to understanding aerodyanmics, atmospheric sciences, thermodynamics, etc. While the one-dimensional fluid flows are rather well understood, the general theory for multi-dimensional flows is comparatively mathematically underdeveloped. The proposers will collaborate in a three year effort to advance the mathematical understanding of the multi-dimensional equations of inviscid compressible fluid dynamics. Success in this project will advance knowledge of this fundamental area of mathematics and mechanics and will introduce a new generation of researchers to the outstanding problems in the field.