# The pressureless limits of Riemann solutions to the Euler equations of one-dimensional compressible fluid flow with a source term

## TL;DR

This paper investigates the limiting behavior of Riemann solutions to inhomogeneous Euler equations as the adiabatic exponent approaches one, revealing convergence to pressureless systems with delta shocks or contact discontinuities, supported by numerical validation.

## Contribution

It rigorously characterizes the limits of Riemann solutions for inhomogeneous Euler equations as gamma approaches one, including the formation of delta shocks and vacuum states, with numerical confirmation.

## Key findings

- Two-shock solutions tend to delta shocks with weighted delta measures.
- Two-rarefaction solutions tend to contact discontinuities with vacuum states.
- Numerical results support the theoretical convergence analysis.

## Abstract

In this paper, we study the limits of Riemann solutions to the inhomogeneous Euler equations of one-dimensional compressible fluid flow as the adiabatic exponent $\gamma$ tends to one. Different from the homogeneous equations, the Riemann solutions of the inhomogeneous system are non self-similar. It is rigorously shown that, as $\gamma$ tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a source term, and the intermediate density between the two shocks tends to a weighted $\delta$-mesaure which forms the delta shock; while any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution of the pressureless Euler system with a source term, whose intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical results to confirm the theoretical analysis.