Riemann Problems with a Kink

We study the Riemann problem for isothermal ow of a gas in a thin pipe with a kink in it. This is modeled by a 22 system of conservation laws with Dirac measure sink term concentrated at the location of the bends in the pipe. We show that the Riemann problem for this system of equations always has a unique solution, given an extra condition relating the speeds on both sides of the kink. Furthermore, we study the related problem where the ow is perturbed by an continuous addition of momentum at distinct points. Under certain conditions we show that also this Riemann problem has a unique solution. 0. Introduction. We consider the ow of an isothermal gas in a (innnitely) long thin pipe of constant cross section. If the walls of the pipe have no eeect on the ow, and the pipe is straight, this can be modeled by the system of conservation laws ((5], p. 56) (0.1) t + (v) x = 0; (v) t + (v 2 +) x = 0: Here, (x; t) denotes the density of the gas, and v(x; t) the velocity. The position along the pipe is described by the coordinate x, and t denotes the time variable. These equations describe the conservation of mass and momentum, respectively. In this paper we discuss the situation where the pipe is not straight, but has a one or several kinks in it. In between these kinks the pipe is straight. Hence the pipe can be described by a polygonal curve, and we ignore gravity. As in the model without kinks, we let (x; t) denote the density of the gas, and v(x; t) its velocity. We now let x be the arc-length parameter along the pipe, or rather the curve describing the pipe. Away from the kinks, conservation of mass and momentum is given by (0.1). It remains to determine the equations holding at the kinks. Since the cross section of the pipe is assumed to be constant on each side of a kink, conservation of mass reads as before (0.2) t + (v) x = 0: In general, we can not assume that and v are continuous at the location of the kink. Since a kink is always located at the same x, which we for simplicity assume to be at x = 0, discontinuities at kinks must satisfy a Rankine-Hugoniot condition where the …