带有源项的Chaplygin气体非对称Keyfitz-Kranzer方程组含狄拉克初值的广义黎曼问题
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摘要
本文主要研究了带有源项的Chaplygin气体非对称Keyfitz-Kranzer方程组含狄拉克初值的广义黎曼问题.由于非齐次项的影响,带有源项的Chaplygin气体非对称Keyfitz-Kranzer方程组的黎曼解不再是自相似的.我们利用广义Rankine-Hugoniot条件和熵条件,构造性地得到了带有源项的Chaplygin气体非对称Keyfitz-Kranzer方程组含狄拉克初值的整体广义解.
This paper is concerned with the Riemann problem with delta initial data for Chaplygin nonsymmetric Keyfitz-Kranzer system with a Coulomb-like friction term. It is interesting to see that the source term makes the Riemann solutions no longer self-similar. Delta contact discontinuities appear in some situations. Under generalized Rankine-Hugoniot conditions and entropy condition, we obtain the propagation speed, position and strength of delta shock wave. Furthermore, under delta initial data, stability of generalized solutions are obtained.
This paper is concerned with the Riemann problem with delta initial data for Chaplygin nonsymmetric Keyfitz-Kranzer system with a Coulomb-like friction term. It is interesting to see that the source term makes the Riemann solutions no longer self-similar. Delta contact discontinuities appear in some situations. Under generalized Rankine-Hugoniot conditions and entropy condition, we obtain the propagation speed, position and strength of delta shock wave. Furthermore, under delta initial data, stability of generalized solutions are obtained.
引文
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[2]AW A,RASCLE M.Resurrection of “second order” models of traffic flow[J].SIAM J Appl Math,2000,60:918-938.
[3]GUO L H.The Riemann problem of the transport equations for the generalized chaplygin gas[J].Journal of Xinjinag University(Natural Science Edition),2013,30(2):170-176.
[4]SHEN C,SUN M N.Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed AW-Rascle model[J].Journal of Differential Equations,2010,249(12):3024-3051.
[5]SUN M N.Interactions of elementary waves for the AW-Rascle model[J].Siam Journal on Applied Mathematics,2009,69(6):1542-1558.
[6]LU Y G.Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type[J].Funct Anal,2012,261(10):2797-2815.
[7]CHENG H J.Delta shock waves for a linearly degenerate hyperbolic system of conservation laws of Keyfitz-Kranzer type[J].Adv Math Phys,2013,Article ID 958120,10 pages.
[8]GUO L H,YIN G.Limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type[J].The Scientific World Journal,2014,2014(3),Article ID287256,11 pages.
[9]TEMPLE B.Global solution of the Cauchy problem for a class of 2×2 nonstrictly hyperbolic conservation laws[J].Adv in Appl Math,1982,3(3):335-375.
[10]JAMES F,PENG Y J,PERTHAME B.Kinetic formulation for chromatography and some other hyperbolic systems[J].Math Pures Appl,1995,74:367-385.
[11]GUO L H,LI T,PAN L J,et al.The Riemann problem with delta initial data for the one-dimensional chaplygin gas equations with a source term[J].Nonlinear Analysis Real World Applications,2018,41:588-606.
[12]RYKOV E W,YU G,SINAI YA G.Generalized variational principle,global weak solutions and behavior with random initial data for system of conservation laws arising in adhesion particle dynamics[J].Comm Math Phys,1996,177(2):349-380.
[13]WANG Z,ZHANG Q L.The Riemann problem with delta initial data for the one-dimensional chaplygin gas equations[J].Acta Math Sci,2012,32B(3):825-841.
[14]YANG H C,SUN W H.The Riemann problem with initial data for a class of coupled hyperbolic of conservation laws[J].Nonlinear Anal TMA,2007,67(11):3041-3049.
[15]李舒琪.非对称Keyfitz-Kranzer方程组的Riemann问题及其基本波的相互作用[D].乌鲁木齐:新疆大学,2018.Li S Q.The Riemann problem and interaction of elementary waves for the nonsymmetric Keyfitz-Kranzer equations[D].Urumqi:Xinjiang University,2018.
[16]FACCANONI G,MANGENEY A.Exact solution for granular flows[J].Int J Numer Anal Methods Geomech,2013,37(10):1408-1433.
[17]BENTO M C,BERTOLAMI O,SEN A A.Generalized Chaplygin gas,accelerated expansion and dark-energy-matter unification[J].Phys Rev D,2002,66(4):043507.
[18]BILIC N,TUPPER G B,VIOLLIER R D.Dark matter,dark energy and the Chaplygin gas[J].arXiv:astro-ph/0207423.
[19]DEV A,ALCANIZ J S,JAIN D.Cosmological consequences of a Chaplygin gas dark energy[J].Phys Rev D,2003,67:023515.
[20]SETARE M R.Holographic chaplygin DGP cosmologies[J].Internat J Modern Phys D,2009,18(3):419-427.
[21]SHEN C.The Riemann problem for the pressureless Euler system with the coulomb-like friction term[J].IMA Journal of Applied Mathematics,2016,81(1):76-99.
[22]SHEN C.The Riemann problem for the chaplygin gas equations with a source term[J].Z Angew Math Mech,2016,96(6):681-695.
[23]SUN M N.The exact Riemann solutions to the generalized chaplygin gas equations with friction[J].Communications in Nonlinear Science and Numerical Simulation,2016,36:342-353.
[24]SHENG W C,ZHANG T.The Riemann problem for the transportation equations in gas dynamics[J].Mem Amer Math Soc,1999,137(654):1-77.
[25]NEDELJKOV M,OBERGUGGENBERGER M.Interactions of delta shock waves in a strictly hyperbolic system of conservation laws[J].J Math Anal Appl,2008,344(2):1143-1157.
[26]SHEN C,SUN M N.Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws[J].Nonlinear Anal TMA,2010,73(10):3284-3294.