# Kinetic Relaxation to Entropy Based Coupling Conditions for Isentropic Flow on Networks

## TL;DR

This paper establishes the existence of weak solutions for isentropic gas flow on networks using kinetic relaxation methods, introducing new coupling conditions and entropy flux inequalities applicable to complex network junctions.

## Contribution

It introduces a kinetic relaxation approach with entropy-based coupling conditions for isentropic flow on networks, including cases with complex junctions and boundary conditions.

## Key findings

- Existence of weak solutions for isentropic gas flow on networks.
- Development of kinetic coupling functions satisfying entropy inequalities.
- Applicability to networks with multiple junctions and boundary conditions.

## Abstract

We consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions. First, we construct approximate solutions by a vector-valued BGK model with a kinetic coupling function. Introducing so-called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations. We will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution. These inequalities define the macroscopic coupling condition. Our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles. We give several examples for coupling functions and prove corresponding entropy flux inequalities. We prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross-sectional area.