Smooth and discontinuous junctions in the p-system

Consider the p-system describing the subsonic flow of a fluid in a pipe with section a = a ( x ) . We prove that the resulting Cauchy problem generates a Lipschitz semigroup, provided the total variation of the initial datum and the oscillation of a are small. An explicit estimate on the bound of the total variation of a is provided, showing that at lower fluid speeds, higher total variations of a are acceptable. An example shows that the bound on TV ( a ) is mandatory, for otherwise the total variation of the solution may grow arbitrarily.