Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

TL;DR

This paper develops Radon measure solutions for Riemann problems in isentropic compressible Euler equations of polytropic gases, revealing unique delta shocks, infinite solutions with delta shocks, and conditions for their existence, with implications for physical modeling.

Contribution

It introduces the first construction of singular measure solutions with delta shocks for strictly hyperbolic Euler systems of polytropic gases, expanding the understanding of solution behaviors.

Findings

Unique delta shock solution under entropy conditions

Infinite solutions involving delta shocks and rarefaction waves

Existence of a unique delta shock in generalized Riemann problems

Abstract

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures, and the solutions admit the concentration of mass. It is found that, under the requirement of satisfying the over-compressing entropy condition: (i) there is a unique delta shock solution, corresponding to the case that has two strong classical Lax shocks; (ii) for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave, or two shocks with one being weak, there are infinitely many solutions, each consists of a delta shock and a rarefaction wave; (iii) there is no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves. These solutions are self-similar. Furthermore, for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data, there always exists a unique delta shock for at least a short time. It could be prolonged to a global solution. Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass (particle). Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified. This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases, that is strictly hyperbolic, and whose characteristics are both genuinely nonlinear. We also discuss possible physical interpretations and applications of these new solutions.