THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX*

doi:10.1007/s10473-024-0102-6

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (1): 37-77.doi: 10.1007/s10473-024-0102-6

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX^*

Yinzheng Sun¹, Aifang Qu^1,†, Hairong Yuan²

1. Department of Mathematics, Shanghai Normal University, Shanghai 200234, China;
2. School of Mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

收稿日期:2022-12-11 修回日期:2023-07-24 出版日期:2024-02-25 发布日期:2024-02-27
通讯作者: † Aifang Qu, E-mail: afqu@shnu.edu.cn
作者简介:Hairong Yuan, E-mail: hryuan@math.ecnu.edu.cn; Yinzheng Sun, E-mail: sunyz1742@163.com
基金资助:
National Natural Science Foundation of China (11871218, 12071298), and in part by the Science and Technology Commission of Shanghai Municipality (21JC1402500, 22DZ2229014).

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX^*

Yinzheng Sun¹, Aifang Qu^1,†, Hairong Yuan²

1. Department of Mathematics, Shanghai Normal University, Shanghai 200234, China;
2. School of Mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

Received:2022-12-11 Revised:2023-07-24 Online:2024-02-25 Published:2024-02-27
Contact: † Aifang Qu, E-mail: afqu@shnu.edu.cn
About author:Hairong Yuan, E-mail: hryuan@math.ecnu.edu.cn; Yinzheng Sun, E-mail: sunyz1742@163.com
Supported by:
National Natural Science Foundation of China (11871218, 12071298), and in part by the Science and Technology Commission of Shanghai Municipality (21JC1402500, 22DZ2229014).

摘要/Abstract

摘要： We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux, more specifically, for pressureless flow on the left and polytropic flow on the right separated by a discontinuity $x=x(t)$. We prove that this problem admits global Radon measure solutions for all kinds of initial data. The over-compressing condition on the discontinuity $x=x(t)$ is not enough to ensure the uniqueness of the solution. However, there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve $x=x(t)+0$, in addition to the full adhesion condition on its left-side. As an application, we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas. In particular, we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas. This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

关键词: compressible Euler equations, Riemann problem, Radon measure solution, delta shock, discontinuous flux, wave interactions

Abstract: We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux, more specifically, for pressureless flow on the left and polytropic flow on the right separated by a discontinuity $x=x(t)$. We prove that this problem admits global Radon measure solutions for all kinds of initial data. The over-compressing condition on the discontinuity $x=x(t)$ is not enough to ensure the uniqueness of the solution. However, there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve $x=x(t)+0$, in addition to the full adhesion condition on its left-side. As an application, we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas. In particular, we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas. This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

Key words: compressible Euler equations, Riemann problem, Radon measure solution, delta shock, discontinuous flux, wave interactions

中图分类号:

35L65

Yinzheng Sun, Aifang Qu, Hairong Yuan. THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX^*[J]. 数学物理学报(英文版), 2024, 44(1): 37-77.

Yinzheng Sun, Aifang Qu, Hairong Yuan. THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX^*[J]. Acta mathematica scientia,Series B, 2024, 44(1): 37-77.

参考文献

[1] Adimurthi, Jaffr$\acute{e}$ J, Gowda G. Godunov-type methods for conservation laws with a flux function discontinuous in space. SIAM J Numer Anal, 2004, 42(1): 179-208
[2] Adimurthi, Siddhartha M, Gowda G. Optimal entropy solutions for conservation laws with discontinuous flux-functions. J Hyperbolic Differ Equ, 2005, 2(4): 783-837
[3] Aekta A, Manas S, Abhrojyoti S, Ganesh V. Solutions with concentration for conservation laws with discontinuous flux and its applications to numerical schemes for hyperbolic systems. Stud Appl Math, 2020, 145(2): 247-290
[4] Andreianov B. New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux. ESAIM: Proceedings and Surveys, 2015, 50: 40-65
[5] Andreianov B, Karlsen K, Risebro N. A theory of $L^{1}$-dissipative solvers for scalar conservation laws with discontinuous flux. Arch Rational Mech Anal, 2011, 201(1): 27-86
[6] Berthelin F, Vovelle J. A Bhatnagar-Gross-Krook approximation to scalar conservation laws with discontinuous flux. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2010, 140(5): 953-972
[7] Darko M. New entropy conditions for scalar conservation laws with discontinuous flux. Discrete and Continuous Dynamical Systems, 2011, 30(4): 1191-1210
[8] Ding M, Li Y. An overview of piston problems in fluid dynamics//Hyperbolic Conservation Laws and Related Analysis with Applications, Springer Proc Math Stat. Heidelberg: Springer, 2014: 161-191
[9] Fazio R, LeVeque R. Moving mesh methods for one-dimensional hyperbolic problems using CLAWPACK. Comp Math Appl, 2003, 45(1): 273-298
[10] Gao L, Qu A, Yuan H. Delta shock as free piston in pressureless Euler flows. Z Angew Math Phys, 2022, 73(3): Art 114
[11] Gimse T, Risebro N. Solution of Cauchy problem for a conservation law with discontinuous flux function. SIAM J Math Anal, 1992, 23(3): 635-648
[12] Guerra G, Shen W. The Cauchy problem for a non strictly hyperbolic $3\times3$ system of conservation laws arising in polymer flooding. Commun Math Sci, 2021, 19(6): 1491-1507
[13] Guerra G, Shen W. Vanishing viscosity and backward Euler approximations for conservation laws with discontinuous fux. SIAM J Math Anal, 2019, 51(4): 3112-3144
[14] Jin Y, Qu A, Yuan H. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Commun Pure Appl Anal, 2021, 20(7): 2665-2685
[15] Jin Y, Qu A, Yuan H. Radon measure solutions to Riemann problems for isentropic compressible Euler equations of polytropic gases. Commun Appl Math Comput, 2023, 7(3): 1097-1129
[16] LeFloch P, Thanh M. The Riemann problem for the shallow water equations with discontinuous topography. Commun Math Sci, 2007, 5(4): 865-885
[17] LeVeque R. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press, 2002
[18] Li J, Sheng W, Zhang T, Zheng Y. Two-dimensional Riemann problems: from scalar conservation laws to compressible Euler equations. Acta Mathematica Scientia, 2009, 29B(4): 777-802
[19] Li T, Yu W.Boundary Value Problems for Quasilinear Hyperbolic Systems. Durham: Duke University Math Series V, 1985
[20] Liu T. The free piston problem for gas dynamics. J Differ Equ, 1978, 30(2): 175-191
[21] Liu T, Smoller J. On the vacuum state for the isentropic gas dynamics equations. Adv Appl Math, 1980, 1(4): 345-359
[22] Nedeljkov M. Shadow waves: entropies and interactions for delta and singular shocks. Arch Rational Mech Anal, 2010, 197(2): 489-537
[23] Qu A, Yuan H. Measure solutions of one-dimensional piston problem for compressible Euler equations of Chaplygin gas. J Math Anal Appl, 2019, 481(1): 123486
[24] Qu A, Yuan H. Radon measure solutions for steady compressible Euler equations of hypersonic-limit conical flows and Newton's sine-squared law. J Differ Equ, 2020, 269(1): 495-522
[25] Qu A, Yuan H, Zhao Q. Hypersonic limit of two-dimensional steady compressible Euler flows passing a straight wedge. Z Angew Math Mech, 2020, 100(3): e201800225
[26] Qu A, Yuan H, Zhao Q. High Mach number limit of one-dimensional piston problem for non-isentropic compressible Euler equations: polytropic gas. J Math Phys, 2020, 61(1): 011507
[27] Shen W. On the Cauchy problems for polymer flooding with gravitation. J Differ Equ, 2016, 261(1): 627-653
[28] Takeno S. Free piston problem for isentropic gas dynamics. Japan J Indust Appl Math, 1995, 12(2): 163-194
[29] Towers J. Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J Numer Anal, 2000, 38(2): 681-698

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX^*

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX^*

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