ONE-DIMENSIONAL HYDRODYNAMIC FLOW IN COMPLEX NETWORKS AND SOME GENERALIZATIONS

The one-dimensional hydrodynamic flow in complex networks that include many branches, junction points, and open boundary points is studied. The shallow-water equations are linearized, and the compatibility conditions at junction points and the boundary conditions at some or all of the open boundary points are analysed. Existence and uniqueness of smooth solutions is discussed and, under certain assumptions, proved. The necessary compatibility conditions depend on whether the flow is subcritical or super-critical. The results are extended to linear hyperbolic systems of more than two equations in networks. In particular, the flow over a network of channels with mobile (erodible) beds, modelled by means of a system of three equations, is analysed.