Coherence and chattering of a one-way valve

In this paper we analyze the effects of a one-way valve on the isothermal gas flow through a pipe. The valve keeps the flow at a constant value * > 0, if possible; otherwise it is closed. First, for fixed *, we define a Riemann solver and characterize the coherence of its initial data; coherence is a necessary condition for the construction of solutions to a general initial-value problem based on a wave-front tracking scheme. We also give an example of an invariant and coherent domain where the valve can be either open or closed. Second, for suitable compact sets of initial data we make precise the range of values * that guarantee the coherence. At last, in the case of a real valve with finite reaction time, we show the chattering (rapid switch on and off) of the valve in correspondence with incoherent initial data.