# Existence theory for well-balanced Euler model

## TL;DR

This paper proves the existence of globally-in-time weak solutions with bounded total variation for a class of Euler equations with source terms, using a well-balanced Glimm method that preserves equilibria.

## Contribution

It introduces a novel application of the well-balanced Glimm method to establish global existence of solutions for Euler models with source terms.

## Key findings

- Existence of global weak solutions with bounded total variation.
- Construction of approximate solutions converging to the exact solution.
- Preservation of fluid equilibria using the well-balanced method.

## Abstract

We study the initial value problem for a kind of Euler equation with a source term. Our main result is the existence of a globally-in-time weak solution whose total variation is bounded on the the domain of definition, allowing the existence of shock waves. Our proof relies on a well-balanced random choice method called Glimm method which preserves the fluid equilibria and we construct a sequence of approximate weak solutions which converges to the exact weak solution of the initial value problem, based on the construction of exact solutions of the generalized Riemann problem associated with initially piecewise steady state solutions.