Communications on Applied Mathematics and Computation >

2023 , Vol. 5 >Issue 3: 1097 - 1129

DOI: https://doi.org/10.1007/s42967-022-00187-0

ORIGINAL PAPER

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

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  • 1. Center for Partial Differential Equations, School of Mathematical Sciences, East China Normal University, Shanghai, 200241, China;
    2. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China;
    3. School of Mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, 200241, China

Received date: 2021-11-05

  Revised date: 2022-01-23

  Online published: 2023-08-29

Supported by

This work was supported by the National Natural Science Foundation of China under Grants No. 11871218, No. 12071298, and by the Science and Technology Commission of Shanghai Municipality under Grant No. 18dz2271000. The authors were grateful to Professor Jiequan Li for his valuable comments on a draft of this paper, and particularly the observation on connections between delta shocks and free pistons, in a private conversation. Aifang Qu appreciated very much the support and the hospitality of the IMS, during her visit at the Chinese University of Hong Kong.

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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 are 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.

Key words： Compressible Euler equations; Radon measure solution; Delta shock; Riemann problem; Non-uniqueness

Cite this article

Yunjuan Jin, Aifang Qu, Hairong Yuan . Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases[J]. Communications on Applied Mathematics and Computation, 2023 , 5(3) : 1097 -1129 . DOI: 10.1007/s42967-022-00187-0

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