Irrotational flows for Chaplygin gas. Conical waves

Remark that real flows should be non-isentropic. The constancy of the entropy is not compatible once shock waves develop. It is only expected that the amplitude of oscillations for the entropy are of cubic order with respect to those of the flow. There is one notable exception, that of a Chaplygin gas, whose pressure has the form p(ρ, s) = g(s)− f(s)/ρ ; then the entropy remains constant even beyond shock formation, if it was at initial time. This is proved in [5] for d = 2; see [6] for d = 3.