非等熵Chaplygin气体测度值解存在性

Abstract

Abstract:

In this paper, we consider the Riemann problem of one-dimensional non-isentropic Chaplygin gas dynamics. In the case that the pressure and internal energy are general, we construct the classical solution by the characteristic analysis under a sufficient and necessary condition on the Riemann data. As the density ρ concentrates, the δ shock waves exist. According to the theory of Radon measure, the general Rankine-Hugoniot is induced. Combining with entropy condition, we obtain the measure solution for this problem. This extends the result for the isentropic Chaplygin gas.

Key words: Non-isentropic Chaplygin gas, Riemann problem, Measure solution, General Rankine-Hugoniot condition

CLC Number:

O175

Yufeng Chen,Tingting Chen,Zhen Wang. The Existence of the Measure Solution for the Non-Isentropic Chaplygin Gas[J].Acta mathematica scientia,Series A, 2020, 40(4): 833-841.

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References 16

1	Serre D . Multidimensional shock interaction for a Chaplygin gas. Arch Ration Mech Anal, 2009, 191 (3): 539- 577 doi: 10.1007/s00205-008-0110-z
2	Chaplygin S A . On gas jets. Scientific Memoirs Moscow Univ Math Phys, 1902, 21, 1- 121
3	Tsien H S . Two-dimensional subsonic flow of compressible fluids. Aeronaut Sci, 1939, 6 (10): 399- 407 doi: 10.2514/8.916
4	Gorini V, Kamenshcik A, Moschella U, Pasquier V. The Chaplygin gas as a model for dark energy. 2004, arXiv: gr-qc/0403062
5	Setare M R . Holographic Chaplygin DGP cosmologies. Internat J Modern Phys D, 2009, 18 (3): 419- 427 doi: 10.1142/S0218271809014558
6	Brenier Y . Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations. J Math Fluid Mech, 2005, 7, 326- 331 doi: 10.1007/s00021-005-0162-x
7	Wang Z , Zhang Q . The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations. Acta Math Sci, 2012, 32B (3): 825- 841
8	Qu A , Wang Z . Stability of the Riemann solutions for a Chaplygin gas. J Math Anal Appl, 2014, 409 (1): 347- 361 doi: 10.1016/j.jmaa.2013.07.018
9	Pang Y . Delta shock wave in the compressible Euler equations for a Chaplygin gas. J Math Anal Appl, 2017, 448, 245- 261 doi: 10.1016/j.jmaa.2016.10.078
10	屈爱芳, 王丽. 非等熵Chaplygin气体极限黎曼解关于扰动的依赖性. 数学年刊A辑, 2018, 39 (2): 219- 228
	Qu A F , Wang L . The dependence of Limit Riemann solution for a non-isentropic Chaplygin gas on perturbation. Chinese Journal of Contemporaty Mathematics, 2018, 39 (2): 219- 228
11	朱荔, 盛万成. 绝热流Chaplygin气体的Riemann问题. 应用数学与计算数学学报, 2010, 24 (1): 9- 16 doi: 10.3969/j.issn.1006-6330.2010.01.002
	Zhu L , Sheng W . The Riemann problem of adiabatic Chaplygin gas dynamic system. Comm Appl Math Comput, 2010, 24 (1): 9- 16 doi: 10.3969/j.issn.1006-6330.2010.01.002
12	Cheng H . Riemann problem for one-dimensional system of conservation laws of mass, momentum and energy in zeropressure gas dynamics. Differ Equ Appl, 2012, 4, 653- 664
13	Cheng H , Yang H . Delta shock waves in chromatography equation. Math Anal Appl, 2011, 380, 475- 485 doi: 10.1016/j.jmaa.2011.04.002
14	Yang H , Zhang Y . Delta shock waves with Dirac delta function in both components for systems of conservation laws. J Differential Equations, 2014, 257 (12): 4369- 4402 doi: 10.1016/j.jde.2014.08.009
15	Qu A, Yuan H, Zhao Q. High Mach number limit of one-dimensional piston problem for non-isentropic compressible Euler equations: polytropic gas. 2019, arXiv: 1904.03362
16	Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag, 1994

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The Existence of the Measure Solution for the Non-Isentropic Chaplygin Gas

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