A note on the Riemann solutions to the isentropic Euler equations in the vanishing pressure limit

The behaviour of the solutions to the Riemann problem for the isentropic Euler equations when the pressure vanishes is analysed. It is shown that any solution composed of a 1-shock wave and a 2-rarefaction wave tends to a two-shock wave when the pressure gets smaller than a fixed value determined by the Riemann data; by contrast, any solution composed of a 1-rarefaction wave and a 2-shock wave tends to a two-rarefaction wave. The two situations are illustrated with numerical tests.