Concentration and Cavitation in the Vanishing Pressure Limit of Solutions to a Simplified Isentropic Relativistic Euler Equations

We identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions to a simplified isentropic relativistic Euler equations. Firstly, both the explicit expressions and geometric properties of the rarefaction wave curve and shock wave curve based on any left state are given with the help of Lorentz invariance, and the Riemann problem for this system is considered. Then, we rigorously prove that, as pressure vanishes, any two-shock Riemann solution tends to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between the two shocks tends to a weighted δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-measure, which forms the delta shock wave. This describes the phenomenon of mass concentration. On the other hand, any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations and the nonvacuum intermediate state in between tends to a vacuum state, which reveals the phenomenon of cavitation. Both concentration and cavitation are fundamental and physical in fluid dynamics.