# The Cauchy problem for the standard one pressure system of two fluid flows with energy equations

## TL;DR

This paper rigorously constructs approximate solutions to a two-fluid flow system with energy equations, proving convergence to Radon measures and providing a numerical method that aligns with previous solutions.

## Contribution

It introduces a mathematically rigorous approach to approximate and analyze solutions of the two-fluid flow system, linking PDEs to ODEs for numerical computation.

## Key findings

- Radon measures approximate solutions match previous numerical results
- The method provides a convergent numerical scheme for the system
- Numerical experiments confirm the theoretical convergence and accuracy

## Abstract

We construct with full rigorous mathematical proof a family of approximate solutions to the Cauchy problem for the standard system of two fluid flows with energy equations and we pass to the limit by weak compactness to obtain Radon measures that satisfy the equations in a natural weak sense. Our method provides a convergent numerical method for the numerical calculation of these Radon measures by reducing the system of partial differential equations in the case of these approximate solutions to a system of ordinary differential equations. We observe numerically on the standard Toumi shock tube problem that the Radon measures from our method agree with the numerical solutions previously obtained by other authors with various different numerical methods. In a subsequent numerical paper, using a standard confident scheme with splittings and vanishing viscosity (independent on the above construction), we observe exactly the numerical solution given by our mathematical proof.