# Coupling conditions for the 3x3 Euler system

## TL;DR

This paper extends analytical properties of the 2x2 fluid flow equations to the full 3x3 Euler system in ducts with varying sections, focusing on coupling conditions at junctions and the importance of bounded total variation.

## Contribution

It introduces coupling conditions for the 3x3 Euler system in pipe networks and demonstrates the necessity of bounded total variation for analytical properties.

## Key findings

- Established coupling conditions for the 3x3 Euler system.
- Proved the necessity of bounded total variation of pipe sections.
- Extended results from simple to complex pipe configurations.

## Abstract

This paper is devoted to the extension to the full $3\times3$ Euler system of the basic analytical properties of the equations governing a fluid flowing in a duct with varying section. First, we consider the Cauchy problem for a pipeline consisting of 2 ducts joined at a junction. Then, this result is extended to more complex pipes. A key assumption in these theorems is the boundedness of the total variation of the pipe's section. We provide explicit examples to show that this bound is necessary.