Note on the compressible euler equations with zero temperature

where p(t, z), ~(2, t) are the mean density and velocity of the flow, p(z, t) is the pressure. Assume the gas under consideration is polytropic. Then the temperature, density, and pressure are related with RT = p/p where R is a positive constant proportional to the molecular weight of gas and assumed to be unit for simplicity. So p = Tp. Roughly speaking, the pressure p vanishes as T goes to zero. This note is just, to investigate the behavior of solution (1) as the temperature T drops to zero by the simple Riemann problem. The Riemann problem to (1) is well known, subject to the initial data