The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up

The one-dimensional isothermal Euler equations are a well-known model for the flow of gas through a pipe. An essential part of the model is the source term that models the influence of gravity and friction on the flow. In general the solutions of hyperbolic balance laws can blow-up in finite time. We show the existence of initial data with arbitrarily large C1-norm of the logarithmic derivative where no blow up in finite time occurs. The proof is based upon the explicit construction of product solutions. Often it is desirable to have such analytical solutions for a system described by partial differential equations, for example to validate numerical algorithms, to improve the understanding of the system and to study the effect of simplifications of the model. We present solutions of different types: In the first type of solutions, both the flow rate and the density are increasing functions of time. We also present a second type of solutions where on a certain time interval, both the flow rate and the pressure decrease. In pipeline networks, the bi-directional use of the pipelines is sometimes desirable. In this paper we present a classical solution of the isothermal Euler equations where the direction of the gas flow changes. In the solution, at the time before the direction of the flow is reversed, the gas flow rate is zero everywhere in the pipe.