用分离的Delta函数法研究非对称Keyfitz-Kranzer系统中Delta激波的交互性

Abstract

Abstract: In this paper, we study the interactions of delta shock waves with contact discontinuities for the nonsymmetric Keyfitz-Kranzer system with split delta functions. The solutions are obtained constructively when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter ε→0.

Key words: Nonsymmetric Keyfitz-Kranzer system, Riemann problem, Delta shock wave, Wave interaction

CLC Number:

O175.29

Li Huahui, Shao Zhiqiang. Interactions of Delta Shock Waves for the Nonsymmetric Keyfitz-Kranzer System with Split Delta Functions[J].Acta mathematica scientia,Series A, 2017, 37(4): 714-729.

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References

[1] Aw A, Rascle M. Resurrection of "second order" models of traffic flow. SIAM J Appl Math, 2000, 60:916-938
[2] Bedran M L, Soares V, Aráujo M E. Temperature evolution of the FRW universe filled with modified Chaplygin gas. Phys Lett B, 2008, 659:462-465
[3] Bilic N, Tupper G B, Viollier R. Dark matter, dark energy and the Chaplygin gas. 2002, arXiv:astroph/0207423
[4] Chaplygin S. On gas jets. Sci Mem Moscow Univ Math Phys, 1904, 21:1-121
[5] Chen G Q, Liu H. Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J Math Anal, 2003, 34:925-938
[6] Chen G Q, Liu H. Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. Physica D, 2004, 189:141-165
[7] Cheng H J. Delta shock waves for a linearly degenerate hyperbolic system of conservation laws of KeyfitzKranzer type. Advances in Mathematical Physics, 2013, Article ID:958120
[8] Danilov V G, Shelkovich V M. Delta-shock waves type solution of hyperbolic systems of conservation laws. Quart Appl Math, 2005, 63:401-427
[9] Gorini V, Kamenshchik A, Moschella U, Pasquier V. The Chaplygin gas as a model for dark energy. 2004, arXiv:gr-qc/0403062
[10] Guo L H, Zhang Y, Yin G. Interactions of delta shock waves for the Chaplygin gas equations with spilt delta functions. J Math Anal Appl, 2014, 410:190-201
[11] Guo L H, Zhang Y, Yin G. Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions. Math Meth Appl Sci, 2015, 38:2131-2148
[12] 黄梅香. 几类非线性双曲守恒律方程组的Riemann问题[D]. 福州:福州大学, 2016 Huang M X. Riemann Problems of Several Classes of Nonlinear Hyperbolic Conservation Laws Equations[D]. Fuzhou:Fuzhou University, 2016
[13] Keyfitz B L, Kranzer H C. A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch Rational Mech Anal, 1980, 72:219-241
[14] Korchinski D J. Solutions of a Riemann Problem for 2×a System of Conservation Laws Possessing no Classical Weak Solution[D]. New York:Adelphi University, 1977
[15] Li J Q. Note on the compressible Euler equations with zero temperature. Appl Math Lett, 2001, 14:519-523
[16] Lu Y. Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type. J Funct Anal, 2011, 261:2792-2815
[17] Nedeljkov M. Singular shock waves in interactions. Quart Appl Math, 2008, 66:281-302
[18] Nedeljkov M. Delta and singular delta locus for one dimensional systems of conservation laws. Math Methods Appl Sci, 2004, 27:931-955
[19] Nedeljkov M, Oberguggenberger M. Interactions of delta shock waves in a strictly hyperbolic system of conservation laws. J Math Anal Appl, 2008, 344:1143-1157
[20] Setare M R. Holographic Chaplygin gas model. Phys Lett B, 2007, 648:329-332
[21] Shao Z Q, Huang M X. Interactions of delta shock waves for the Aw-Rascle traffic model with split delta functions. J Appl Anal Comput, 2017, 7:119-133
[22] Shelkovich V M. δ-and δ'-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes. Russian Math Surveys, 2008, 63:473-546
[23] Shen C, Sun M. Interactions of delta shock waves for the transport equations with split delta functions. J Math Anal Appl, 2009, 351:747-755
[24] Shen C, Sun M. Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws. Nonlinear Anal TMA, 2010, 73:3284-3294
[25] Shen C, Sun M. Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model. J Differential Equations, 2010, 249:3024-3051
[26] Sheng W C, Zhang T. The Riemann Problem for the Transportation Equations in Gas Dynamics. Providence, RI:Amer Math Soc, 1999
[27] Tan D, Zhang T, Zheng Y. Delta shock waves as limits of vanishing viscosity for hyperbolic system of conservation laws. J Differential Equations, 1994, 112:1-32
[28] Tsien H S. Two dimensional subsonic flow of compressible fluids. J Aeron Sci, 1939, 6:393-407

Interactions of Delta Shock Waves for the Nonsymmetric Keyfitz-Kranzer System with Split Delta Functions

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