vanishing viscosity, networks. This work has received funding from the Alexander von Humboldt-Professorship program, the Transregio 154 Project "Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks" of the DFG, the grant PID2020-112617GB-C22 of MINECO (Spain), and the COST Action grant CA18232, "Mathematical models for interacting dynamics on networks" (MAT-DYN-NET). J. A. B<acute accent>arcena-Petisco is funded by the Grant PID2021-126813NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe", and by the grant IT1615-22 funded by the Basque Government. M. Cavalcante has been partially funded by CAPES-MATHAMSUD 88887.368708/2019. G. M. Coclite and N. De Nitti are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit & aacute; e le loro Applicazioni (GNAMPA) of the 2, Investment 1.4 (Call for tender No. 3138 of 16/12/2021), of Italian Ministry of University and