球对称 Chaplygin 气体相对论 Euler 方程组的奇性形成

数学物理学报 ›› 2023, Vol. 43 ›› Issue (2): 481-490.

球对称 Chaplygin 气体相对论 Euler 方程组的奇性形成

侍迎春(),赖耕^*()

上海大学数学系上海 200444

收稿日期:2022-05-21 修回日期:2022-10-17 出版日期:2023-04-26 发布日期:2023-04-17
通讯作者: 赖耕, E-mail: laigeng@shu.edu.cn
作者简介:侍迎春, E-mail: shiyc@shu.edu.cn
基金资助:
国家自然科学基金(12071278)

Formation of Singularities in Solutions to Spherically Symmetric Relativistic Euler Equations for a Chaplygin Gas

Shi Yingchun(),Lai Geng()

Department of Mathematics, Shanghai University, Shanghai 200444

Received:2022-05-21 Revised:2022-10-17 Online:2023-04-26 Published:2023-04-17
Supported by:
NSFC(12071278)

摘要/Abstract

摘要：

该文研究了具有 Chaplygin 气体状态方程的相对论 Euler 方程组经典解的奇性形成. 给出了关于初值的一个充分条件, 使得 Chaplygin 气体相对论方程组的一维 Cauchy 问题经典解的质能密度 $\rho$ 本身在有限时间内发生破裂.

关键词: 相对论Euler方程组, Chaplygin 气体, 奇性形成, 特征分解

Abstract:

This paper studies the formation of singularities in smooth solutions to three-dimensional (3D) spherically symmetric relativistic Euler matrixs with a Chaplygin gas matrix of state. We give a sufficient condition on the initial data to obtain that the mass-energy density itself of the classical solutions to the Cauchy problem blows up in finite time.

Key words: Relativistic Euler matrixs, Chaplygin gas, Singularities formation, Characteristic decomposition

中图分类号:

O175.27

侍迎春, 赖耕. 球对称 Chaplygin 气体相对论 Euler 方程组的奇性形成[J]. 数学物理学报, 2023, 43(2): 481-490.

Shi Yingchun, Lai Geng. Formation of Singularities in Solutions to Spherically Symmetric Relativistic Euler Equations for a Chaplygin Gas[J]. Acta mathematica scientia,Series A, 2023, 43(2): 481-490.

参考文献 33

[1]	Landau L D, Lifschitz E M. Fluid Mechanics. Oxford: Pergamon, 1987
[2]	Chaplygin S. On gas jets. Sci Mem Moscow Univ Math Phys, 1904, 21: 1-121
[3]	Tsien H S. Two dimensional subsonic flow of compressible fluids. J Aeronaut Sci, 1939, 6: 399-407 doi: 10.2514/8.916
[4]	Karman T V. Compressibility effects in aerodynamics. J Aeronaut Sci, 1941, 8: 337-365 doi: 10.2514/8.10737
[5]	Bento M C, Bertolami O, Sen A A. Generalized Chalplygin gas, accelerated expansion and dark-energy-matter unification. Phys Rev D, 2002, 66: 043507
[6]	Cruz N, Lepe S, Pena F. Dissipative generalized Chaplygin gas as phantom dark energy physics. Phys Lett B, 2007, 646: 177-182 doi: 10.1016/j.physletb.2006.12.070
[7]	Setare M R. Holographic Chaplygin gas model. Phys Lett B, 2007, 648: 329-332 doi: 10.1016/j.physletb.2007.03.025
[8]	Setare M R. Interacting holographic generalized Chaplygin gas model. Phys Lett B, 2007, 654: 1-6 doi: 10.1016/j.physletb.2007.08.038
[9]	Brenier Y. Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas matrixs. J Math Fluid Mech, 2005, 7: 326-331
[10]	Cheng H J, Yang H C. Riemann problem for the relativistic Chaplygin Euler matrixs. J Math Anal Appl, 2011, 381: 17-26 doi: 10.1016/j.jmaa.2011.04.017
[11]	Cheng H J, Yang H C. Riemann problem for the isentropic relativistic Chaplygin Euler matrixs. Z Angew Math Phys, 2012, 63: 429-440 doi: 10.1007/s00033-012-0199-7
[12]	Godin P. Global existence of a class of smooth 3D spherically symmetric flows of Chaplygin gases with variable entropy. J Math Pures Appl, 2007, 87: 91-117 doi: 10.1016/j.matpur.2006.10.011
[13]	Ding B B, Witt I, Yin H C. The global smooth symmetric solution to 2-D full compressible Euler system of Chaplygin gases. J Differential Equations, 2015, 258: 445-482 doi: 10.1016/j.jde.2014.09.018
[14]	Hou F, Yin H C. Global smooth axisymmetric solutions to 2D compressible Euler matrixs of Chaplygin gases with non-zero vorticity. J Differential Equations, 2019, 267: 3114-3161 doi: 10.1016/j.jde.2019.03.038
[15]	Hou F, Yin H C. On global axisymmetric solutions to 2D compressible full Euler matrixs of Chaplygin gases. Discrete Contin Dyn Syst, 2010, 40: 1435-1492 doi: 10.3934/dcds.2020083
[16]	Kong D X, Liu K, Wang Y. Global existence of smooth solutions to two-dimensional compressible isentropic Euler matrixs for Chaplygin gases. Sci China Math, 2010, 53: 719-738 doi: 10.1007/s11425-010-0060-4
[17]	Lei Z, Wei C H. Global radial solutions to 3D relativistic Euler matrixs for non-isentropic Chaplygin gases. Math Ann, 2017, 367: 1363-1401 doi: 10.1007/s00208-016-1396-z
[18]	Alinhac S. Une solution approchée en grand temps des équations d'Euler compressibles axisymétriques en dimen-sion deux. Commun Partial Differ Equ, 1992, 17: 447-490
[19]	Alinhac S. Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux. Invent Math, 1993, 111: 627-670 doi: 10.1007/BF01231301
[20]	Athanasiou N, Zhu S G. Formation of singularities for the relativistic Euler matrixs. J Differential Equations, 2021, 284: 284-317 doi: 10.1016/j.jde.2021.03.010
[21]	Chen G. Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler matrixs. J Hyperbolic Differ Equ, 2011, 8: 671-690 doi: 10.1142/S0219891611002536
[22]	Chen G, Pan R H, Zhu S G. Singularity formation for the compressible Euler matrixs. SIAM J Math Anal, 2017, 49: 2591-2614 doi: 10.1137/16M1062818
[23]	Christodoulou D. The formation of Shocks in 3-dimensional Fluids. European Mathematical Society, 2007
[24]	Christodoulou D, Miao S. Compressible Flow and Euler's Equations. Boston, Beijing: International Press; Higher Education Press, 2014
[25]	Guo Y, Tahvildar Z S. Formation of singularities in relativistic fluid dynamics and in spherically symmetric plasma dynamics. Cntemp Math, 1999, 238: 151-161
[26]	Luk J, Speck J. Shock formation in solutions to the 2D compressible Euler matrixs in the presence of non-zero vorticity. Invent Math, 2018, 214: 1-169 doi: 10.1007/s00222-018-0799-8
[27]	Pan R H, Smoller J. Blowup of smooth solutions for relativistic Euler matrixs. Commun Math Phys, 2006, 262: 729-755 doi: 10.1007/s00220-005-1464-9
[28]	Majda A. Compressible fluid flow and systems of conservation laws in several space variables. Applied Mathematical Sciences, 1984, 53
[29]	Qu P. Mechanism of singularity formation for quasilinear hyperbolic systems with linearly degenerate characteristic fields. J Differential Equations, 2011, 251: 2066-2081 doi: 10.1016/j.jde.2011.07.005
[30]	Lai G, Zhu M. Formation of singularities of solutions to the compressible Euler matrixs for a Chaplygin gas. Applied Mathematics Letters, 2022, 129: 107978
[31]	Li J Q, Zhang T, Zheng Y X. Simple waves and a characteristic decomposition of the two dimensional compressible Euler matrixs. Comm Math Phys, 2006, 267: 1-12 doi: 10.1007/s00220-006-0033-1
[32]	Li J Q, Zheng Y X. Interaction of rarefaction waves of the two-dimensional self-similar Euler matrixs, Arch Ration Mech Anal, 2009, 193: 623-657 doi: 10.1007/s00205-008-0140-6
[33]	Li T T, Yu W C. Boundary Value Problem for Quasilinear Hyperbolic Systems. Duke University, 1985