THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES

doi:10.1007/s10473-020-0505-y

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (5): 1229-1239.doi: 10.1007/s10473-020-0505-y

THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES

蒋伟峰¹, 王振²

1. College of Science, China Jiliang University, Hangzhou 310018, China;
2. College of Science, Wuhan University of Technology, Wuhan 430071, China

收稿日期:2019-08-24 修回日期:2020-05-13 出版日期:2020-10-25 发布日期:2020-11-04
通讯作者: Zhen WANG E-mail:zwang@whut.edu.cn
作者简介:Weifeng JIANG,E-mail:casujiang89@gmail.com
基金资助:
The first author was supported by the Natural Science Foundation of Zhejiang (LQ18A010004), the second author was supported by the Fundamental Research Funds for the Central Universities (WUT: 2020IB011).

THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES

Weifeng JIANG¹, Zhen WANG²

1. College of Science, China Jiliang University, Hangzhou 310018, China;
2. College of Science, Wuhan University of Technology, Wuhan 430071, China

Received:2019-08-24 Revised:2020-05-13 Online:2020-10-25 Published:2020-11-04
Contact: Zhen WANG E-mail:zwang@whut.edu.cn
Supported by:
The first author was supported by the Natural Science Foundation of Zhejiang (LQ18A010004), the second author was supported by the Fundamental Research Funds for the Central Universities (WUT: 2020IB011).

摘要/Abstract

摘要： In this article, by the mean-integral of the conserved quantity, we prove that the one-dimensional non-isentropic gas dynamic equations in an ideal gas state do not possess a bounded invariant region. Moreover, we obtain a necessary condition on the state equations for the existence of an invariant region for a non-isentropic process. Finally, we provide a mathematical example showing that with a special state equation, a bounded invariant region for the non-isentropic process may exist.

关键词: Euler equations, gas dynamic, non-isentropic, existence of invariant region

Abstract: In this article, by the mean-integral of the conserved quantity, we prove that the one-dimensional non-isentropic gas dynamic equations in an ideal gas state do not possess a bounded invariant region. Moreover, we obtain a necessary condition on the state equations for the existence of an invariant region for a non-isentropic process. Finally, we provide a mathematical example showing that with a special state equation, a bounded invariant region for the non-isentropic process may exist.

Key words: Euler equations, gas dynamic, non-isentropic, existence of invariant region

中图分类号:

35L65

蒋伟峰, 王振. THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES[J]. 数学物理学报(英文版), 2020, 40(5): 1229-1239.

Weifeng JIANG, Zhen WANG. THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES[J]. Acta mathematica scientia,Series B, 2020, 40(5): 1229-1239.

参考文献 17

[1]	Tartar L. The compensated compactness method applied to systems of conservation laws//Syst Nonlin Part Differ Equ. Springer, 1983:263-285
[2]	DiPerna R J. Convergence of the viscosity method for isentropic gas dynamics. Commun Math Phys, 1983, 91:1-30
[3]	DiPerna R J. Convergence of approximate solutions to conservation laws. Arch Ration Mech Anal, 1983, 82:27-70
[4]	Ding X X, Chen G Q, Luo P Z. A supplement to the papers convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (Ⅱ)-(Ⅲ). Acta Math Sci, 1989, 9(1):43-44
[5]	Ding X X, Chen G Q, Luo P Z. Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Commun Math Phys Anal, 1989, 121:63-84
[6]	Barles G, Souganidis P E. Convergence of approximation schemes for fully nonlinear second order equations. Asymototic Anal, 1991, 4(3):271-283
[7]	Lions P L, Perthame B, Souganidis P E. Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Commun Pure Appl Math, 1996, 49(6):599-638
[8]	Lions P L, Perthame B, Tadmor E. Kinetic formulation of the isentropic gas dynamics and p-systems. Commun Math Phys Anal, 1994, 163(2):415-431
[9]	Huang F M, Wang Z. Convergence of viscosity solutions for isothermal gas dynamics. SIAM J Math Anal, 2002, 34(3):595-610
[10]	Li J Q, Sheng W C, Zhang T, et al. Two-dimensional Riemann problems:from scalar conservation laws to compressible Euler equations. Acta Math Sci, 2009, 29(4):777-802
[11]	Lu Y G. Convergence of the viscosity method for a nonstrictly hyperbolic system. Acta Math Sci, 1992, 12(2):230-239
[12]	Frid H. Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws. Discrete Contin Dyn Syst, 1995, 1(4):585-593
[13]	Frid H. Maps of convex sets and invariant regions for finite-difference systems of conservation laws. Arch Ration Mech Anal, 2001, 160(3):245-269
[14]	Tadmor E. A minimum entropy principle in the gas dynamics equations. Appl Numer Math, 1986, 2(3/5):211-219
[15]	Chueh K N, Conley C C, Smoller J A. Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ Math J, 1977, 26(2):373-392
[16]	Jiang W F, Wang Z. The invariant region for the equations of nonisentropic gas dynamics. ANZIAM J, 2017, 58(3/4):428-435
[17]	Smoller J. Shock Waves and Reaction-Diffusion Equations. Springer, 2012

THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES

THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献 17

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	孙庆有, 陆云光, Christian KLINGENBERG. GLOBAL L^∞ SOLUTIONS TO SYSTEM OF ISENTROPIC GAS DYNAMICS IN A DIVERGENT NOZZLE WITH FRICTION[J]. 数学物理学报(英文版), 2019, 39(5): 1213-1218.
[2]	袁海荣. TIME-PERIODIC ISENTROPIC SUPERSONIC EULER FLOWS IN ONE-DIMENSIONAL DUCTS DRIVING BY PERIODIC BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2019, 39(2): 403-412.
[3]	陈贵强, Matthew RIGBY. STABILITY OF STEADY MULTI-WAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW[J]. 数学物理学报(英文版), 2018, 38(5): 1485-1514.
[4]	侯飞. ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE[J]. 数学物理学报(英文版), 2017, 37(4): 949-964.
[5]	施小丁, 雍燕, 张英龙. VANISHING VISCOSITY FOR NON-ISENTROPIC GAS DYNAMICS WITH INTERACTING SHOCKS[J]. 数学物理学报(英文版), 2016, 36(6): 1699-1720.
[6]	曹文涛, 黄飞敏, 李天虹, 于慧敏. GLOBAL ENTROPY SOLUTIONS TO AN INHOMOGENEOUS ISENTROPIC COMPRESSIBLE EULER SYSTEM[J]. 数学物理学报(英文版), 2016, 36(4): 1215-1224.
[7]	陈宇, 周忆. SIMPLE WAVES OF THE TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS[J]. 数学物理学报(英文版), 2015, 35(4): 855-875.
[8]	王术, 徐自立. INCOMPRESSIBLE LIMIT OF THE NON-ISENTROPIC MAGNETOHYDRODYNAMIC EQUATIONS IN BOUNDED DOMAINS[J]. 数学物理学报(英文版), 2015, 35(3): 719-745.
[9]	许秋菊\|蔡虹\|谭忠. TIME PERIODIC SOLUTIONS OF NON-ISENTROPIC COMPRESSIBLE MAGNETOHYDRODYNAMIC SYSTEM[J]. 数学物理学报(英文版), 2015, 35(1): 216-234.
[10]	张映辉, 吴国春. GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR THE 3D COMPRESSIBLE NON–ISENTROPIC EULER EQUATIONS WITH DAMPING[J]. 数学物理学报(英文版), 2014, 34(2): 424-434.
[11]	Geng Chen, Robin Young. THE VACUUM IN NONISENTROPIC GAS DYNAMICS[J]. 数学物理学报(英文版), 2012, 32(1): 339-351.
[12]	肖玲, 李海梁, 杨彤, 邹晨. COMPRESSIBLE NON-ISENTROPIC BIPOLAR NAVIER-STOKES-POISSON SYSTEM IN R³[J]. 数学物理学报(英文版), 2011, 31(6): 2169-2194.
[13]	K.T. Joseph and Manas R. Sahoo. SOME EXACT SOLUTIONS OF 3-DIMENSIONAL ZERO-PRESSURE GAS DYNAMICS SYSTEM[J]. 数学物理学报(英文版), 2011, 31(6): 2107-2121.
[14]	Maria Schonbek. NONEXISTENCE OF PSEUDO-SELF-SIMILAR SOLUTIONS TO INCOMPRESSIBLE EULER EQUATIONS[J]. 数学物理学报(英文版), 2011, 31(6): 2305-2312.
[15]	谢华朝, 訾瑞昭. REMARKS ON THE NONLINEAR INSTABILITY OF INCOMPRESSIBLE EULER EQUATIONS[J]. 数学物理学报(英文版), 2011, 31(5): 1877-1888.