气体动力学中Chaplygin气体的Riemann问题
|
摘要
本文研究了气体动力学中Chaplygin气体的Riemann问题。
第二章首先介绍了关于双曲守恒律系统的一些基本概念。然后对一维和二维双曲守恒律方程的一般性理论分别给出了介绍。
第三章研究了一维Chaplygin气体的Riemann问题。和多方气体不同的是,Chaplygin气体的所有特征都是线性退化的。所以Chaplygin气体的基本波都是接触间断。我们把相平面分为五个区域,有趣的是其中有一个区域会出现delta波。delta波由广义Rankine-Hugoniot关系式的提出,并对delta波从速度,位置和权上分别给出了精确的描述,并且给出了delta波出现的熵条件。这些解展示了宇宙演化过程中的某些现象,例如黑洞。
第四章研究了四片常状态初值的两维Chanplygin气体的Riemann问题。初值在原点发出的四条射线上间断,每条初始间断线在t＞0时发射出三道平面基本波((?)或(?))+J+((?)或(?))。这十二道波在(x,y,t)空间中将在以原点为顶点的锥体中相互作用。为了使问题简化而又不失实质,引进假设:每条初始间断线在t＞0时只发射出一道平面基本波。于是,问题就被简化成四道基本波的相互作用。根据四道波的不同组合,将问题分成十四类。其中有六类是无旋的。除了2J~++2J~-以外,利用广义特征分析方法逐个进行讨论,构造出了超音区的解.有趣的是有些情况会出现delta波。亚音区的边界包括音速线和滑移面。这些解也展示了宇宙演化过程中的某些现象,例如黑洞。
第五章研究了两维Chaplygin气体具有轴对称性质的解。利用轴对称和自相似假设,我们可以把偏微分方程组的初值问题化为非自治常微分方程组的无穷远边值问题。奇点对应于四维相空间中的两维流形,并且对应于物理空间中特征曲面。全局解至少包含三段非平凡的轨道连接。与多方气体不同的是,即使轴向速度大于零的时候也会出现间断;轴向速度小于零时会出现质量集中的现象。这些解展示了宇宙演化的某些现象,例如黑洞的形成和演化,宇宙的暴涨和膨胀。
In this article,we study the Riemann problem for Chaplygin gas of conservation laws.
In section 2,we introduces some useful concepts for the hyperbolic system firstly. Then some general theories about one-dimensional and two-dimensional hyperbolic systems are introduced in the next part of this section,respectively.
In section 3,we discuss the one one-dimensional Riemann problem of Chaplygin gas. Different from polytropic gas,the eigenvalues of Chaplygin gas are linear degenerate.So their elementary waves are contact discontinuities.We divided the phase plane into five regions.More interesting thing is that there appear delta wave in one region.Due to the introduction of a suitable generalized Rankine-Hugoniot relation,delta shock gets a well depiction from velocity,location and weight.And we give the entropy condition about the delta wave.The solutions exhibit some phenomena,such as black hole,in the evolution of universe.
In section 4,we discuss the initial data of Riemanna problem with four constant states.There are four jumps,each of them emits three waves:((?) or(?)),J~±,((?) or(?)) at t>0.These elementary waves interaction in the cone which vertex is orign in space (x,y,t).The problem is how the twelve waves interact and match together ultimately. Obviously,it is too complicated to deal with but its key point is the interaction of different elementary waves,so we made a restriction that each jump at infinity emits exactly one elementary wave.According to combinations of the four waves and compatibility,We classified these problems into 14 cases.Six of them are irrotational.By using method of generalized characteristic analysis,we construct supersonic solution for each case except the case of 2J~+ + 2J~-.More interesting thing is that delta waves may appear in some cases.And Dirichlet boundary value problems in subsonic domain are formed for some cases.The boundaries of the domains composed of sonic curves and slip lines,he solutions exhibit some phenomena,such as black hole,in the evolution of universe.
In section 5,we discuss axisymmetric solutions for the two-dimensional Chaplygin gas.We usethe axisymmetry and self-similarity assumptions to reduce the initial-value problem of partial differential equations to a infinite boundary-value problem for a system of non-autonomous ordinary differential equations.Singularity points of the system con-sist of two-dimensional manifolds in the four-dimensional phase space.These singularity points correspond to surface of characteristics in physical space-time.A global solution may consist of as many as three nontrivial connecting orbits chained together.Different from polytropic gas,discontinuities exist even though the velocity of radially direction is positive and if the velocity of radially direction is negative,there appear concentration phenomenon.The solutions exhibit some phenomena,such as black hole formation and development,expansion and explosive expansion,in the evolution of universe.
第二章首先介绍了关于双曲守恒律系统的一些基本概念。然后对一维和二维双曲守恒律方程的一般性理论分别给出了介绍。
第三章研究了一维Chaplygin气体的Riemann问题。和多方气体不同的是,Chaplygin气体的所有特征都是线性退化的。所以Chaplygin气体的基本波都是接触间断。我们把相平面分为五个区域,有趣的是其中有一个区域会出现delta波。delta波由广义Rankine-Hugoniot关系式的提出,并对delta波从速度,位置和权上分别给出了精确的描述,并且给出了delta波出现的熵条件。这些解展示了宇宙演化过程中的某些现象,例如黑洞。
第四章研究了四片常状态初值的两维Chanplygin气体的Riemann问题。初值在原点发出的四条射线上间断,每条初始间断线在t＞0时发射出三道平面基本波((?)或(?))+J+((?)或(?))。这十二道波在(x,y,t)空间中将在以原点为顶点的锥体中相互作用。为了使问题简化而又不失实质,引进假设:每条初始间断线在t＞0时只发射出一道平面基本波。于是,问题就被简化成四道基本波的相互作用。根据四道波的不同组合,将问题分成十四类。其中有六类是无旋的。除了2J~++2J~-以外,利用广义特征分析方法逐个进行讨论,构造出了超音区的解.有趣的是有些情况会出现delta波。亚音区的边界包括音速线和滑移面。这些解也展示了宇宙演化过程中的某些现象,例如黑洞。
第五章研究了两维Chaplygin气体具有轴对称性质的解。利用轴对称和自相似假设,我们可以把偏微分方程组的初值问题化为非自治常微分方程组的无穷远边值问题。奇点对应于四维相空间中的两维流形,并且对应于物理空间中特征曲面。全局解至少包含三段非平凡的轨道连接。与多方气体不同的是,即使轴向速度大于零的时候也会出现间断;轴向速度小于零时会出现质量集中的现象。这些解展示了宇宙演化的某些现象,例如黑洞的形成和演化,宇宙的暴涨和膨胀。
In this article,we study the Riemann problem for Chaplygin gas of conservation laws.
In section 2,we introduces some useful concepts for the hyperbolic system firstly. Then some general theories about one-dimensional and two-dimensional hyperbolic systems are introduced in the next part of this section,respectively.
In section 3,we discuss the one one-dimensional Riemann problem of Chaplygin gas. Different from polytropic gas,the eigenvalues of Chaplygin gas are linear degenerate.So their elementary waves are contact discontinuities.We divided the phase plane into five regions.More interesting thing is that there appear delta wave in one region.Due to the introduction of a suitable generalized Rankine-Hugoniot relation,delta shock gets a well depiction from velocity,location and weight.And we give the entropy condition about the delta wave.The solutions exhibit some phenomena,such as black hole,in the evolution of universe.
In section 4,we discuss the initial data of Riemanna problem with four constant states.There are four jumps,each of them emits three waves:((?) or(?)),J~±,((?) or(?)) at t>0.These elementary waves interaction in the cone which vertex is orign in space (x,y,t).The problem is how the twelve waves interact and match together ultimately. Obviously,it is too complicated to deal with but its key point is the interaction of different elementary waves,so we made a restriction that each jump at infinity emits exactly one elementary wave.According to combinations of the four waves and compatibility,We classified these problems into 14 cases.Six of them are irrotational.By using method of generalized characteristic analysis,we construct supersonic solution for each case except the case of 2J~+ + 2J~-.More interesting thing is that delta waves may appear in some cases.And Dirichlet boundary value problems in subsonic domain are formed for some cases.The boundaries of the domains composed of sonic curves and slip lines,he solutions exhibit some phenomena,such as black hole,in the evolution of universe.
In section 5,we discuss axisymmetric solutions for the two-dimensional Chaplygin gas.We usethe axisymmetry and self-similarity assumptions to reduce the initial-value problem of partial differential equations to a infinite boundary-value problem for a system of non-autonomous ordinary differential equations.Singularity points of the system con-sist of two-dimensional manifolds in the four-dimensional phase space.These singularity points correspond to surface of characteristics in physical space-time.A global solution may consist of as many as three nontrivial connecting orbits chained together.Different from polytropic gas,discontinuities exist even though the velocity of radially direction is positive and if the velocity of radially direction is negative,there appear concentration phenomenon.The solutions exhibit some phenomena,such as black hole formation and development,expansion and explosive expansion,in the evolution of universe.
引文
[1]M.L.Bedran,V.Soares and M.E.Araujo.Temperature evolution of the FRW universe filled with modified Chaplygin gas,Phys.Lett.B,659(2008),462-465.
[2]N.Bilic,G.B.Tupper and R.Viollier.Dark matter,dark energy and the Chaplygin gas,astroph /0207423.
[3]Y.Brenier,Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations,J.Math.Fluid Mech.,7(2005),S326-S331.
[4]J.M.Burgers,A mathematical model illustrating the theory of turbulence,Advances in Applied Mechanics,1(1948),171-199.
[5]J.Carr,Applications of Centre Manifold Theory,Applied Mathematical Sciences,Sringer-Verlag,Newyork,Berlin,Heidelberg,35(1981).
[6]T.Chang and G.Q.Chen,Some fundamental concepts about system of two spatial dimensional conservation laws,Acta Math.Sci,6(1986),463-474.
[7]T.Chang,G.Chen and S.Yang,On the Riemann problem for 2-D compressible Euler equations I.Interaction of shocks and rarefaction waves,Discrete and Continuous Dynamical Systems,4(1995),555-584.
[8]T.Chang and L.Hsiao,The Riemann Problem and Interaction of Waves in Gas Dynamics,in:Pitman Monographs and Surveys in Pure and Applied Mathematics,Vol.41,Longman Scientific and Technical,Essex,Englang,1989.
[9]S.Chaplygin,On gas jets,Sci.Mem.Moscow Univ.Math.Phys.,21(1904),1-121.
[10]G.Q.Chen,Remarks on spherically symmetric solution of the compressible Euler equation,Proceedings of the Royal Society of Edinburgh,127(1997),243-259.
[11]G.Q.Cheng,D.N.Li and D.C.Tan,Structure of Riemann solution for 2-dimmensional scalar conservation laws,J.Differentional Equations,127(1)(1996),124-147.
[12]G.Q.Chen,H.L.Liu,Formation of Delta-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to the Isentropic Euler Equations,SIAM J.Mathematical Analysis,34(2003),925-938.
[13]S.X.Chen,A singular multi-dimensional piston problem in compressible flow,J.Diff.Equations,189(2003),292-317.
[14]S.X.Chen,Z.J.Wang and Y.Q.Zhang,Global existence of shock front solutions to the axially symmetric piston problem for compressible fluids,J.Hyper.Diff.Equations,1(1)(2001),51-84.
[15]Norman Cruz,Samuel Lepe and Francisco Pe(n|¨)a,Dissipative generalized Chaplygin gas as phantom dark energy Physics,Phys.Lett.,B646(2007),177-182.
[16]E.Conway and J.Smoler,Global solution of the Cauchy problem for quasilinear first order equations in several variables,Comm.Pure Appl.Math.,ⅩⅠⅩ(1966),95-105.
[17]M.Crandall,H.Ishii and P.L.Lions,User's guide to viscosity solutions of second order partial differential equations,Bulletin of the AMS,27(1992),1-68.
[18]R.Courant and K.O.Friedrichs,Supersonic Flow and Shock Waves,New York:Wiley 1948.
[19]R.Courant and D.Hilbert,Methods of Mathematical Physics,Vol.2,Wiley-Interscience,New York.
[20]C.M.Dafermos,Hyperbolic conservation laws in Continuum Physics,Springer,New York,1999.
[21]Z.Dai and T.Zhang,Existence of a global smooth solution for a degenerate Coursat problem of gas dynamics.Arch.Bat.Mech.Anal.,155(2000),277-298.
[22]A.Davidson,D.Karasik,and Y.Lederer,Cold Dark Matter from Dark Energy,gr-qc/0111107,(2001).
[23]V.Gorini,A.Kamenshchik,Moschella,The chaplygin gas as an model for dark energy,grqc /0403062,23(2004).
[24]J.Guckenheimer,Shocks and rarefactions in two space dimensions,Arch.Rat.Mech.Anal.,59(1988),281-291.
[25]D.Henry,Geometric theory of semilinear parabolic Equations,Springer Lectures Notes in Mathematics,vol.840.Springer-Ver;ag,New York,Heidelberg,Berlin,1981.
[26]黄飞敏,杨小舟,一类双曲守恒律方程组的二维问题,应用数学学报,21(2)(1998),193-204.
[27]E.Hopf.The partial differential equation u_t+uu_x=μu_(xx).Comm.Pure Appl.Math.,3(1950),201-230.
[28]D.Hoff and J.Smoller,SOlutions in the large for certain nonlinear parabolic systems,Analyse Non Linear,2(1985),213-235.
[29]D.Hoff and J.Smoller,Global existence for systems of parabolic conservation laws in several space variables,J.Differential Equations,68(1987),210-220.
[30]L.Hsiao,T.Luo and T.Yang,Global BV solutions of compressible Euler equatons with spherical symmetry and damping,J.Diff.Equations,146(1998),277-298.
[31]J.K.Hunter,Transverse diffraction of nonlinear waves and singular rays,SIAM J.Appl.Math.,48(1988) 1-37.
[32]K.T.Joseph,A Riemann problem shose viscosity solutions contain delta-measures,Asymptotic Analysis,7(1993),105-120.
[33]A.Kamenshchik,U.Moschella,V.Pasquier,An alternative to quintessence,Phys.Lett.B,511(2001),265-268.
[34]S.Kasuya,Difficulty of a spinning complex scalar field to be dark energy,Phys.Lett.B,515(2001),121-124.
[35]Kelley,A.:The stable,center stable,center,center unstable and unstable manifolds,J.Diff.Eqs.,3(1967) 546-570.
[36]D.X.Kong,Q.Zhang and Q.Zhou,The dynamics of relativistic strings moving in the Minkowski space R~(1+n),Commun.Math.Phys.,269(2007),153-174
[37]S.N.Kruzkov,Order quasilinear equations with several independent variables,Math.USSR Sb.,10(1970),271-243.
[38]P.D.Lax,Hyperbolic system of conservation laws Ⅱ,Comm.Pure Appl.Math.10(1957),537-566.
[39]P.D.Lax,Hyperblic systems of conservation laws and the mathematical theory of shock waves,SIAM,Philadelphia,(1973).
[40]L.E.Levine,The expansion of a wedge of gas into a vacuum,Math.Proc.Cambridge Philos.Soc.,64(1968),1151-1163.
[41]李大潜,秦铁虎,物理学与偏微分方程,北京:高等教育出版社,(1997).
[42]J.Q.Li,On the two-dimensional gas expansion for compressible Euler equations.SIAM J.Appl.Math.,62(2001),831-852.
[43]J.Q.Li,T.Zhang,S.Yang,The two-dimensional Riemann problem in gas dynamics,in:Pitman Monographs and Surveys in Pure and Applied Mathematics 98,Longman,Scientific and Technical,Harlow,1998.
[44]J.Li,T.Zhang and Y.Zheng,Simple wave and a characteristic decomposition of the two dimensional compressible Euler equations.Communication Math Physics,267(2006),1-12.
[45]J.Q.Li and Y.Zheng,Interaction of rarefaction waves of the two-dimensional self-similar Euler equations,to appear in Arch.Rational Mech.Anal.
[46]Tatsian Li and Wancheng Sheng,The general Riemann problem for the linearized system of two-dimensional isentropic flow in gas dynarnics,Journal of Mathematical Analysis and Applications,276(2)(2002),598-610.
[47]李维新,一维不定常流与冲击波,国防工业出版社,(2003).
[48]W.B.Lindquist,Scalar Riemann problem in two spatial dimensions;piecewise smoothness of solutions and its breakdown,SIAM J.Math.Ana.,17(1986),1178-1197.
[49]T.P.Liu,Admissible solutions of hyperbolic conservation laws,Memoris Amer.Math.Soc.,240(1981).
[50]A.G.Mackie,Two-dimensional quasi-stationary flows in gas dynamics,Proc.Cambridge Philos.Soc.,68(1968),1099-1108.
[51]A.J.Majda,compressible fluid flow and systems of conservation laws in several space variables,Appl.Math.Sciences,53,Springer-Verlag.
[52]Michael S.Turner,Dark energy and the new cosmology,astro-ph/0108103.
[53]T.Nishida and J.Smoller,A class of convergent finite-difference schemes for certain nonlinear parabolic systems,Comm.Pure Appl.Math.,36(1983),785-808.
[54]Oleinik O.,Discontinuous solutons of nonlinear differential equatons,Usp.Mat.Nauk.(N.S.),12,(1957),3-73;English transl,in Amer.Math.Soc.Transl.,Ser.2,26,95-172.
[55]Oleinik O.,On the uniqueness of the generalized solution of the Cauchy problem for a nonlinear system of equations occurring in mechanics.Usp.Mat.Nauk.(N.S.),12(1957),169-176.
[56]B.Riemann,Uber die forpflanzung ebener luftwellen von endlicher schwingungsweite,Abhandl.Koenig.Gesell.Wiss,Goettingen,8(1860):43.
[57]C.W.Schulz-Rinne,Classification of the Riemann problem for two-dimensional gas dynamics.SIAM J.Math.Anal.,24(1993),76-88.
[58]D.Serre,Systems of conservation laws Ⅰ:Hyperbolicity,Entropy,Shock waves;Ⅱ:Geometric structures,Oscillations,and Mixed problems,Cambridge Univ.Press,Cambridge,2000.
[59]D.Serre,Multidimensional shock interaction for a Chaplygin gas,Arch.Rational Mech.Anal.,191(2009),539-577.
[60]M.R.Setare,Holographic Chaplygin gas model,Phys.Lett.B,648(2007),329-332.
[61]M.R.Setare,Interacting holographic generalized Chaplygin gas model,Phys.Lett.B,654(2007),1-6.
[62]单鹏,多维气体动力学基础,北京:北京航空航天大学出版社,(2004).
[63]W.C.Sheng,Two-Dimensional Riemann Problem for Scalar Conservation Laws,J.Differential Equations,183(2002),239-261.
[64]W.C.Sheng,T.Zhang,The Riemann problem for the transportation equations in gas dynamics,Memoirs of the American Mathematical Society,Vol.137,Num.654,1999.
[65]W.C.Sheng,T.Zhang,A Cartoon for the Climbing Ramp Problem of a Shock and von Neumann Paradox,Arch.Rational Mech.Anal.84(2),2007,243-255.
[66]R.Smith,The Riemann problem in gas dynamics,Trans.Amer.Math.Soc.,249(1979).
[67]J.Smoller,Shock Waves and Reaction Diffusion Equations,Berlin,Heidelberg,New York:Springer 1983.
[68]D.C.Tan and T.Zhang,Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws(Ⅰ):four-J cases,J.Differential Equations,111(1994),203-254.
[69]D.C.Tan and T.Zhang,Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws(Ⅱ):initial data consists of some rarefaction,J.Differential Equations,111(1994),255-283.
[70]童秉纲,孔祥言,邓国华,多维气体动力学基础,北京:高等教育出版社,(1990).
[71]H.S.Tsien,Two dimensional subsonic flow of compressible fluids,J.Aeron.Sci.,6(1939),399-407.
[72]T.von Karman,Compressibility effects in aerodynamics,J.Aeron.Sci.,8(1941),337-365.
[73]D.Wagner,The Riemann problem in two space dimensions for a singe conservation laws,SIAM J.Math.Anal.,14(1983),534-559.
[74]Z.J.Wang,Local existence of the shock front solution to the axi-symmetrical piston problem in compressible flow,20(4)(2004),589-604.
[75]韦浩,蔡荣根,暗能量理论研究现状概述,科技导报,vol 23,12(2005),28-32.
[76]H.Yang,Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics,J.Math.Anal.and Appl.,260(2001),18-35.
[77]H.Yang and T.Zhang,On two-dimensional gas expansion for pressure-gradient equations of Euler system.J.Math.Anal.Appl.,298(2004),523-537.
[78]应隆安,腾振寰,双曲守恒律方程及其差分方法,科学出版社,北京,(1991).
[79]H.Zhang and W.Sheng,Guckenheimer structure of solution of Riemann problem with four pieces of constants in two space dimensions for scalar conservation laws,Journal of Shanghai University,4(2006),26-28.
[80]P.Zhang and T.Zhang,Generalized characteristic analysis and Guckenheimer structure,J.Differentional Equaitons,152(2)(1999).409-430.
[81]T.Zhang and Y.X.Zheng,Two dimensional Riemann problem for a single conservation laws,Trans.Amer.Math.Soc.,312(1989),589-619.
[82]T.Zhang and Y.X.Zheng,Conjecture on structure of solution of Riemann problem for 2-D gas dynamic systems,SIAM.J.Math.Anal.,3(1990),493-630.
[83]T.Zhang and Y.X.Zheng,Axisymmetric solutions of the Euler equations for polytropic gases,Arch.Rational Mech.Anal,142(1998),253-279.
[84]T.Zhang and Y.X.Zheng,Exact spiral solutions of the two dimensional compressible Euler equations,Discrete and continuous dynamical system,3(1997),117-133.
[85]T.Zhang and Y.X.Zheng,Explicit and exact swirling solutions of the 2-D Euler equations,Z.angew.Math.Mech.,76(1996),Special issue 2,Applied Analysis,145-148.
[86]张新民,第三讲宇宙中的暗物质和暗能量,物理,vol.34,6(2005).
[87]Y.X.Zheng,Two-dimensional regular reflection for the pressure gradient equation,Acta Mathematicae Applicatae Sinica Vol.22(2006),No.2,pp.177-210(English series).
[88]Y.X.Zheng,Systems of conservation laws.Two-dimensional Riemann problems,Birkh(a|¨)user,Boston,(2001).
[89]Y.X.Zheng,A global solution to a two-dimensional Riemann problem involving shocks as free boundaries.Acta Math.Appl.Sin.,19(2003),559-572.
[90]Y.X.Zheng,The Compressible Euler System in Two Dimensions,Lecture Notes of 2007 Shanghai Mathematics Summer School.International Press and Higher Education Press,in press.
[2]N.Bilic,G.B.Tupper and R.Viollier.Dark matter,dark energy and the Chaplygin gas,astroph /0207423.
[3]Y.Brenier,Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations,J.Math.Fluid Mech.,7(2005),S326-S331.
[4]J.M.Burgers,A mathematical model illustrating the theory of turbulence,Advances in Applied Mechanics,1(1948),171-199.
[5]J.Carr,Applications of Centre Manifold Theory,Applied Mathematical Sciences,Sringer-Verlag,Newyork,Berlin,Heidelberg,35(1981).
[6]T.Chang and G.Q.Chen,Some fundamental concepts about system of two spatial dimensional conservation laws,Acta Math.Sci,6(1986),463-474.
[7]T.Chang,G.Chen and S.Yang,On the Riemann problem for 2-D compressible Euler equations I.Interaction of shocks and rarefaction waves,Discrete and Continuous Dynamical Systems,4(1995),555-584.
[8]T.Chang and L.Hsiao,The Riemann Problem and Interaction of Waves in Gas Dynamics,in:Pitman Monographs and Surveys in Pure and Applied Mathematics,Vol.41,Longman Scientific and Technical,Essex,Englang,1989.
[9]S.Chaplygin,On gas jets,Sci.Mem.Moscow Univ.Math.Phys.,21(1904),1-121.
[10]G.Q.Chen,Remarks on spherically symmetric solution of the compressible Euler equation,Proceedings of the Royal Society of Edinburgh,127(1997),243-259.
[11]G.Q.Cheng,D.N.Li and D.C.Tan,Structure of Riemann solution for 2-dimmensional scalar conservation laws,J.Differentional Equations,127(1)(1996),124-147.
[12]G.Q.Chen,H.L.Liu,Formation of Delta-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to the Isentropic Euler Equations,SIAM J.Mathematical Analysis,34(2003),925-938.
[13]S.X.Chen,A singular multi-dimensional piston problem in compressible flow,J.Diff.Equations,189(2003),292-317.
[14]S.X.Chen,Z.J.Wang and Y.Q.Zhang,Global existence of shock front solutions to the axially symmetric piston problem for compressible fluids,J.Hyper.Diff.Equations,1(1)(2001),51-84.
[15]Norman Cruz,Samuel Lepe and Francisco Pe(n|¨)a,Dissipative generalized Chaplygin gas as phantom dark energy Physics,Phys.Lett.,B646(2007),177-182.
[16]E.Conway and J.Smoler,Global solution of the Cauchy problem for quasilinear first order equations in several variables,Comm.Pure Appl.Math.,ⅩⅠⅩ(1966),95-105.
[17]M.Crandall,H.Ishii and P.L.Lions,User's guide to viscosity solutions of second order partial differential equations,Bulletin of the AMS,27(1992),1-68.
[18]R.Courant and K.O.Friedrichs,Supersonic Flow and Shock Waves,New York:Wiley 1948.
[19]R.Courant and D.Hilbert,Methods of Mathematical Physics,Vol.2,Wiley-Interscience,New York.
[20]C.M.Dafermos,Hyperbolic conservation laws in Continuum Physics,Springer,New York,1999.
[21]Z.Dai and T.Zhang,Existence of a global smooth solution for a degenerate Coursat problem of gas dynamics.Arch.Bat.Mech.Anal.,155(2000),277-298.
[22]A.Davidson,D.Karasik,and Y.Lederer,Cold Dark Matter from Dark Energy,gr-qc/0111107,(2001).
[23]V.Gorini,A.Kamenshchik,Moschella,The chaplygin gas as an model for dark energy,grqc /0403062,23(2004).
[24]J.Guckenheimer,Shocks and rarefactions in two space dimensions,Arch.Rat.Mech.Anal.,59(1988),281-291.
[25]D.Henry,Geometric theory of semilinear parabolic Equations,Springer Lectures Notes in Mathematics,vol.840.Springer-Ver;ag,New York,Heidelberg,Berlin,1981.
[26]黄飞敏,杨小舟,一类双曲守恒律方程组的二维问题,应用数学学报,21(2)(1998),193-204.
[27]E.Hopf.The partial differential equation u_t+uu_x=μu_(xx).Comm.Pure Appl.Math.,3(1950),201-230.
[28]D.Hoff and J.Smoller,SOlutions in the large for certain nonlinear parabolic systems,Analyse Non Linear,2(1985),213-235.
[29]D.Hoff and J.Smoller,Global existence for systems of parabolic conservation laws in several space variables,J.Differential Equations,68(1987),210-220.
[30]L.Hsiao,T.Luo and T.Yang,Global BV solutions of compressible Euler equatons with spherical symmetry and damping,J.Diff.Equations,146(1998),277-298.
[31]J.K.Hunter,Transverse diffraction of nonlinear waves and singular rays,SIAM J.Appl.Math.,48(1988) 1-37.
[32]K.T.Joseph,A Riemann problem shose viscosity solutions contain delta-measures,Asymptotic Analysis,7(1993),105-120.
[33]A.Kamenshchik,U.Moschella,V.Pasquier,An alternative to quintessence,Phys.Lett.B,511(2001),265-268.
[34]S.Kasuya,Difficulty of a spinning complex scalar field to be dark energy,Phys.Lett.B,515(2001),121-124.
[35]Kelley,A.:The stable,center stable,center,center unstable and unstable manifolds,J.Diff.Eqs.,3(1967) 546-570.
[36]D.X.Kong,Q.Zhang and Q.Zhou,The dynamics of relativistic strings moving in the Minkowski space R~(1+n),Commun.Math.Phys.,269(2007),153-174
[37]S.N.Kruzkov,Order quasilinear equations with several independent variables,Math.USSR Sb.,10(1970),271-243.
[38]P.D.Lax,Hyperbolic system of conservation laws Ⅱ,Comm.Pure Appl.Math.10(1957),537-566.
[39]P.D.Lax,Hyperblic systems of conservation laws and the mathematical theory of shock waves,SIAM,Philadelphia,(1973).
[40]L.E.Levine,The expansion of a wedge of gas into a vacuum,Math.Proc.Cambridge Philos.Soc.,64(1968),1151-1163.
[41]李大潜,秦铁虎,物理学与偏微分方程,北京:高等教育出版社,(1997).
[42]J.Q.Li,On the two-dimensional gas expansion for compressible Euler equations.SIAM J.Appl.Math.,62(2001),831-852.
[43]J.Q.Li,T.Zhang,S.Yang,The two-dimensional Riemann problem in gas dynamics,in:Pitman Monographs and Surveys in Pure and Applied Mathematics 98,Longman,Scientific and Technical,Harlow,1998.
[44]J.Li,T.Zhang and Y.Zheng,Simple wave and a characteristic decomposition of the two dimensional compressible Euler equations.Communication Math Physics,267(2006),1-12.
[45]J.Q.Li and Y.Zheng,Interaction of rarefaction waves of the two-dimensional self-similar Euler equations,to appear in Arch.Rational Mech.Anal.
[46]Tatsian Li and Wancheng Sheng,The general Riemann problem for the linearized system of two-dimensional isentropic flow in gas dynarnics,Journal of Mathematical Analysis and Applications,276(2)(2002),598-610.
[47]李维新,一维不定常流与冲击波,国防工业出版社,(2003).
[48]W.B.Lindquist,Scalar Riemann problem in two spatial dimensions;piecewise smoothness of solutions and its breakdown,SIAM J.Math.Ana.,17(1986),1178-1197.
[49]T.P.Liu,Admissible solutions of hyperbolic conservation laws,Memoris Amer.Math.Soc.,240(1981).
[50]A.G.Mackie,Two-dimensional quasi-stationary flows in gas dynamics,Proc.Cambridge Philos.Soc.,68(1968),1099-1108.
[51]A.J.Majda,compressible fluid flow and systems of conservation laws in several space variables,Appl.Math.Sciences,53,Springer-Verlag.
[52]Michael S.Turner,Dark energy and the new cosmology,astro-ph/0108103.
[53]T.Nishida and J.Smoller,A class of convergent finite-difference schemes for certain nonlinear parabolic systems,Comm.Pure Appl.Math.,36(1983),785-808.
[54]Oleinik O.,Discontinuous solutons of nonlinear differential equatons,Usp.Mat.Nauk.(N.S.),12,(1957),3-73;English transl,in Amer.Math.Soc.Transl.,Ser.2,26,95-172.
[55]Oleinik O.,On the uniqueness of the generalized solution of the Cauchy problem for a nonlinear system of equations occurring in mechanics.Usp.Mat.Nauk.(N.S.),12(1957),169-176.
[56]B.Riemann,Uber die forpflanzung ebener luftwellen von endlicher schwingungsweite,Abhandl.Koenig.Gesell.Wiss,Goettingen,8(1860):43.
[57]C.W.Schulz-Rinne,Classification of the Riemann problem for two-dimensional gas dynamics.SIAM J.Math.Anal.,24(1993),76-88.
[58]D.Serre,Systems of conservation laws Ⅰ:Hyperbolicity,Entropy,Shock waves;Ⅱ:Geometric structures,Oscillations,and Mixed problems,Cambridge Univ.Press,Cambridge,2000.
[59]D.Serre,Multidimensional shock interaction for a Chaplygin gas,Arch.Rational Mech.Anal.,191(2009),539-577.
[60]M.R.Setare,Holographic Chaplygin gas model,Phys.Lett.B,648(2007),329-332.
[61]M.R.Setare,Interacting holographic generalized Chaplygin gas model,Phys.Lett.B,654(2007),1-6.
[62]单鹏,多维气体动力学基础,北京:北京航空航天大学出版社,(2004).
[63]W.C.Sheng,Two-Dimensional Riemann Problem for Scalar Conservation Laws,J.Differential Equations,183(2002),239-261.
[64]W.C.Sheng,T.Zhang,The Riemann problem for the transportation equations in gas dynamics,Memoirs of the American Mathematical Society,Vol.137,Num.654,1999.
[65]W.C.Sheng,T.Zhang,A Cartoon for the Climbing Ramp Problem of a Shock and von Neumann Paradox,Arch.Rational Mech.Anal.84(2),2007,243-255.
[66]R.Smith,The Riemann problem in gas dynamics,Trans.Amer.Math.Soc.,249(1979).
[67]J.Smoller,Shock Waves and Reaction Diffusion Equations,Berlin,Heidelberg,New York:Springer 1983.
[68]D.C.Tan and T.Zhang,Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws(Ⅰ):four-J cases,J.Differential Equations,111(1994),203-254.
[69]D.C.Tan and T.Zhang,Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws(Ⅱ):initial data consists of some rarefaction,J.Differential Equations,111(1994),255-283.
[70]童秉纲,孔祥言,邓国华,多维气体动力学基础,北京:高等教育出版社,(1990).
[71]H.S.Tsien,Two dimensional subsonic flow of compressible fluids,J.Aeron.Sci.,6(1939),399-407.
[72]T.von Karman,Compressibility effects in aerodynamics,J.Aeron.Sci.,8(1941),337-365.
[73]D.Wagner,The Riemann problem in two space dimensions for a singe conservation laws,SIAM J.Math.Anal.,14(1983),534-559.
[74]Z.J.Wang,Local existence of the shock front solution to the axi-symmetrical piston problem in compressible flow,20(4)(2004),589-604.
[75]韦浩,蔡荣根,暗能量理论研究现状概述,科技导报,vol 23,12(2005),28-32.
[76]H.Yang,Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics,J.Math.Anal.and Appl.,260(2001),18-35.
[77]H.Yang and T.Zhang,On two-dimensional gas expansion for pressure-gradient equations of Euler system.J.Math.Anal.Appl.,298(2004),523-537.
[78]应隆安,腾振寰,双曲守恒律方程及其差分方法,科学出版社,北京,(1991).
[79]H.Zhang and W.Sheng,Guckenheimer structure of solution of Riemann problem with four pieces of constants in two space dimensions for scalar conservation laws,Journal of Shanghai University,4(2006),26-28.
[80]P.Zhang and T.Zhang,Generalized characteristic analysis and Guckenheimer structure,J.Differentional Equaitons,152(2)(1999).409-430.
[81]T.Zhang and Y.X.Zheng,Two dimensional Riemann problem for a single conservation laws,Trans.Amer.Math.Soc.,312(1989),589-619.
[82]T.Zhang and Y.X.Zheng,Conjecture on structure of solution of Riemann problem for 2-D gas dynamic systems,SIAM.J.Math.Anal.,3(1990),493-630.
[83]T.Zhang and Y.X.Zheng,Axisymmetric solutions of the Euler equations for polytropic gases,Arch.Rational Mech.Anal,142(1998),253-279.
[84]T.Zhang and Y.X.Zheng,Exact spiral solutions of the two dimensional compressible Euler equations,Discrete and continuous dynamical system,3(1997),117-133.
[85]T.Zhang and Y.X.Zheng,Explicit and exact swirling solutions of the 2-D Euler equations,Z.angew.Math.Mech.,76(1996),Special issue 2,Applied Analysis,145-148.
[86]张新民,第三讲宇宙中的暗物质和暗能量,物理,vol.34,6(2005).
[87]Y.X.Zheng,Two-dimensional regular reflection for the pressure gradient equation,Acta Mathematicae Applicatae Sinica Vol.22(2006),No.2,pp.177-210(English series).
[88]Y.X.Zheng,Systems of conservation laws.Two-dimensional Riemann problems,Birkh(a|¨)user,Boston,(2001).
[89]Y.X.Zheng,A global solution to a two-dimensional Riemann problem involving shocks as free boundaries.Acta Math.Appl.Sin.,19(2003),559-572.
[90]Y.X.Zheng,The Compressible Euler System in Two Dimensions,Lecture Notes of 2007 Shanghai Mathematics Summer School.International Press and Higher Education Press,in press.