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.