Euler equations for isentropic gas dynamics with general pressure law

Muhammad Ibrahim,Yu-Pei Lv,Ayman A Aly,Anwarud Din,Hadi Jahanshahi,Abdullahi Yusuf

doi:10.1186/s13662-022-03680-1

Abstract

In this work, we explore the limiting behavior of Riemann solutions to the Euler equations in isentropic gas dynamics with general pressure law. We demonstrate that in the distributional sense the delta wave of zero-pressure gas dynamics is formed by a limit solution. Finally, to present the concentration phenomena, we also offer some numerical outcomes.

Highlights

The primary theme of the paper is the Euler equations of isentropic gas dynamics in Eulerian coordinates, ⎧⎨γτ + (γ v)η = 0, ⎩(γ v)τ + (γ v2 + P)η = 0, (1)with Riemann initial values (γ, v)|τ =0 =⎨(γ, ⎩(γ+, v–), v+), η < 0, η > 0, (2)where v, p(γ ), and γ are the velocity, pressure, and density, respectively, We assume that v– > v+ and the pressure function is
Bouchut [1] proved the presence of the system entropy solution (6) for the Riemann case, and the associated numerical findings affirmed that the alternatives have the concentration of the density, that is, the delta wave
Chen and Lin [19] initially demonstrated the development of vacuum states and δ-shocks of the Riemann solutions to the Euler equations (1) as the stress comes to zero, which describes the concentration and cavitation phenomenon exactly in mathematics

Summary

Introduction

Bouchut [1] proved the presence of the system entropy solution (6) for the Riemann case, and the associated numerical findings affirmed that the alternatives have the concentration of the density, that is, the delta wave. Chen and Lin [19] initially demonstrated the development of vacuum states and δ-shocks of the Riemann solutions to the Euler equations (1) as the stress comes to zero, which describes the concentration and cavitation phenomenon exactly in mathematics. 2, we report some preliminaries on the delta wave solution for the pressure-free gas dynamical system (6).

Results

Conclusion