Full title:  The Pressure Gradient System

Lectures at the Capital Normal University
December 18--31, 2003.

Contents
1. Motivation

  a. Euler system of compressible fluid 1755--59:
     A historical problem.
  b. Compressible fluid are everywhere, e.g., air or
     all matter in the universe in the large scale.
     General applicability.
  c. Usefulness of simpler models: Burgers'.

2. Progress.

  a. One-dimensional system has existence, uniqueness, and
     stability.
  b. Two-dimensional system is now an obvious target.
  c. Models, i.e. ladder problems. One of them is the
     pressure gradient equation

3. Outline

 a. Two derivations of the pressure gradient system
 b. Initial-boundary value problems, Riemann problem,
    Oblique shock reflection, shock hitting a ramp,
    Mach reflections, numerical simulations.
    Ref: Li's book.
 c. Subsonic domain: Existence (new simplified proof)
 d. Supersonic domain: Interaction of two 1-D rarefaction waves;
    Ref: Dai and Zhang
 e. Curved shock waves as free boundary: Ref: AMAS
 f. Regular reflection of oblique shocks, and other fall-outs
    (An announcement.)
 g. Open problems: immediate and intermediate types

 h. Tools: elliptic estimates, tangential oblique derivative
    boundary-value problems. (Book of Chen Yazhe)