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

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