Existence of solution to a model for gas transportation networks on non-flat topography

In this paper we prove the existence of solution to a mathematical model for gas transportation networks on non-flat topography. Firstly, the network topology is represented by a directed graph and then a nonlinear system of numerical equations is introduced whose unknowns are the pressures at the nodes and the mass flow rates at the edges of the graph. This system is written in a compact vector form in terms of the vector of the square pressures at the nodes and then an existence result is proved under some simplifying assumptions. The proof is done in two steps: the first one uses convex analysis tools and the second one the Brouwer fixed-point theorem.