辐射流体力学方程组定解问题适定性的研究
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摘要
此博士论文研究辐射流体力学方程组定解问题的苻干数学理论问题,主要包括一维和高维辐射流体力学方程组初边值问题光滑解的局部存在性,三维辐射流体力学方程组解的奇性以及一维辐射流体力学方程组整体弱解的存在性
辐射流体力学是描述热辐射在流体中的传播以及该辐射对一般流体运动的影响的学科。热辐射在物理问题中的重要性是随着温度的增加而增大的,这是因为辐射的能量密度是随着温度的四次方来变化的。在适当的温度下(比如几千Kelvin),辐射在流体运动过程中所起的作用是通过热辐射过程来传播能量。在更高的温度环境下(比如几百万Kelvin),辐射场的能量和动量密度将控制相应的流体粒子。在这种情况下,辐射场明显地影响流体的运动。因此在高温环境中,研究流体运动就必须考虑辐射的作用。辐射流体力学理论有着广泛的应用,如天体物理,激光核聚变,超新星爆炸理论等。
对于一维和高维辐射流体力学方程组,我们先考虑其初边值问题的适定性。我们将利用Picard迭代,能量估计等方法得到光滑解的局部存在性,唯一性和稳定性。其次对三维等熵辐射流体力学方程组,我们证明该系统的一些C1解无论初值的大小,必在有限时间内产生破裂。最后我们考虑一维辐射流体力学方程组Cauchy问题体熵解。通过补偿列紧的方法,我们得到了在给定任意大的初值下整体的L∞弱熵解的存在性。
Some mathematical problems in radiation hydrodynamics equations are considered in this thesis, including the local existence of smooth solutions to the initial boundary value problem for the one-dimensional and multidimensional radiation hydrodynamics equations, the formation of singularities of solutions to the three-dimensional radiation hydrodynamics equations and the global weak solutions to the one-dimensional radiation hydrodynamics equations.
The radiation hydrodynamics concerns the propagation of thermal radiation through a fluid, and the effect of this radiation on the hydrodynamics describing the fluid motion. The importance of thermal radiation in physical problems increases as the temperature is raised, because the radiation energy density varies as the fourth power of the temperature. At mod-erate temperatures (say, thousands of degrees Kelvin), the role of the radiation is transporting energy by radiative processes. At higher temperatures (say, millions of degrees Kelvin), the energy and momentum densities of the radiation field may become dominating the corre-sponding fluid quantities. In this case, the radiation field significantly affects the dynamics of the fluid. Therefore, at higher temperature condition, one must consider the functions of radiation when we study the motion of fluid. The theory of radiation hydrodynamics owns a wide range of application, including in astrophysical, laser fusion, supernove explosions.
For one-dimensional and multidimensional radiation hydrodynamics equations, we first study the well-posedness of the initial-boundary value problems. By using the Picard iter-ation, energy estimate we obtain the local existence of smooth solutions. Second for the three-dimensional isentropic radiation hydrodynamics equations, we prove that some C1 so-lutions should blow-up in a finite time regardless of the size of the initial disturbance. Finally, for the Cauchy problem of the one-dimensional radiation hydrodynamics equations, by us-ing the compensated compactness argument, we establish the existence of a global entropy solution in L∞with arbitrarily large initial data.
辐射流体力学是描述热辐射在流体中的传播以及该辐射对一般流体运动的影响的学科。热辐射在物理问题中的重要性是随着温度的增加而增大的,这是因为辐射的能量密度是随着温度的四次方来变化的。在适当的温度下(比如几千Kelvin),辐射在流体运动过程中所起的作用是通过热辐射过程来传播能量。在更高的温度环境下(比如几百万Kelvin),辐射场的能量和动量密度将控制相应的流体粒子。在这种情况下,辐射场明显地影响流体的运动。因此在高温环境中,研究流体运动就必须考虑辐射的作用。辐射流体力学理论有着广泛的应用,如天体物理,激光核聚变,超新星爆炸理论等。
对于一维和高维辐射流体力学方程组,我们先考虑其初边值问题的适定性。我们将利用Picard迭代,能量估计等方法得到光滑解的局部存在性,唯一性和稳定性。其次对三维等熵辐射流体力学方程组,我们证明该系统的一些C1解无论初值的大小,必在有限时间内产生破裂。最后我们考虑一维辐射流体力学方程组Cauchy问题体熵解。通过补偿列紧的方法,我们得到了在给定任意大的初值下整体的L∞弱熵解的存在性。
Some mathematical problems in radiation hydrodynamics equations are considered in this thesis, including the local existence of smooth solutions to the initial boundary value problem for the one-dimensional and multidimensional radiation hydrodynamics equations, the formation of singularities of solutions to the three-dimensional radiation hydrodynamics equations and the global weak solutions to the one-dimensional radiation hydrodynamics equations.
The radiation hydrodynamics concerns the propagation of thermal radiation through a fluid, and the effect of this radiation on the hydrodynamics describing the fluid motion. The importance of thermal radiation in physical problems increases as the temperature is raised, because the radiation energy density varies as the fourth power of the temperature. At mod-erate temperatures (say, thousands of degrees Kelvin), the role of the radiation is transporting energy by radiative processes. At higher temperatures (say, millions of degrees Kelvin), the energy and momentum densities of the radiation field may become dominating the corre-sponding fluid quantities. In this case, the radiation field significantly affects the dynamics of the fluid. Therefore, at higher temperature condition, one must consider the functions of radiation when we study the motion of fluid. The theory of radiation hydrodynamics owns a wide range of application, including in astrophysical, laser fusion, supernove explosions.
For one-dimensional and multidimensional radiation hydrodynamics equations, we first study the well-posedness of the initial-boundary value problems. By using the Picard iter-ation, energy estimate we obtain the local existence of smooth solutions. Second for the three-dimensional isentropic radiation hydrodynamics equations, we prove that some C1 so-lutions should blow-up in a finite time regardless of the size of the initial disturbance. Finally, for the Cauchy problem of the one-dimensional radiation hydrodynamics equations, by us-ing the compensated compactness argument, we establish the existence of a global entropy solution in L∞with arbitrarily large initial data.
引文
[1]A. M. Anile, A. M. Blokhin Trakhinin and L. Yu, Investigation of a mathematical model for radiation hydrodynamics, Z Angew Math Phys,50 (1999),677-697.
[2]R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal.,152(4)(2000),327-355.
[3]S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Birkhauser, Boston, (1972).
[4]C. Buet and B. Despres, Asymptotic analysis of fluid models for coupling of radiation and hydronamics, J. of Quantative Spectroscopy and Radiative Transfer,85(2004),385-418.
[5]C. Bardos, F. Golse and D. Levermore, Fluid dynamical limits of kinetic equations Formal derivation,J. Statistical.Phys.,63(1991),323-344;Ⅱ. Convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math.,46(1993),667-753.
[6]C. Bardos, F. Golse and D. Levermore, The acoustic limit of the Boltzmann equation, Arch. Raional Mech. Anal.,153(2000),177-204.
[7]P.L. Bhatnagar, E.P. Gross and M. Krook, A model for collision processes in gases.Ⅰ. small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94(1954),511-525.
[8]L. Boltzmann, Lectures on Gas Theory (Transl. by S. G. Brush), Dover Publ. Inc. New York,1964.
[9]A. Bressan, Hyperbolic Systems of Conservation Laws:The One Dimensional Cauchy Problem, Oxford Univ. Press, Oxford,2000.
[10]J. I. Castor, Radiation Hydrodynamics, Cambridge University Press,2004.
[11]G.-Q. Chen, Convergence of Lax-Friedrichs scheme for isentropic gas dynamics. Ⅲ, Acta Math. Sci.,59(1986),75-120.
[12]G.-Q. Chen, Entropy, Compactness, and Conservation Laws, Lecture Notes, North-western University,2001.
[13]G.-Q. Chen and T. H. Li, Global entropy solution in L∞ to the Euler equations and Euler-Poisson equatins for isothermal fluid with spherical symmetry, Methods and Ap-plications of Analysis.,10(2)(2003),215-244.
[14]G.-Q. Chen and D. Wang, Convergence of shock capturing scheme for the compressible Euler-Poisson equations, Commun. Math. Phys.,179(1996),333-364.
[15]G.-Q. Chen and D. Wang, The Cauchy problem for the Euler equations for compressible fluids, Handbook of Mathematical Fluid Dynamics, Amsterdam:North-Holland,2002, 421-543.
[16]G.-Q. Chen, Y. G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch Rational Mech Anal,179(3)(2008), 369-408.
[17]S. X. Chen, Stability of a Mach configuration, Comm Pure Appl Math,59(2006),1-35.
[18]R. Courant and K. O.Friedrichs, Supersonic Flow and Shock Wave, Springer-Verlag, Berlin, Heidelberg, New York,1976.
[19]R. Courant and D. Hilbert, Methods of Mathematical physics. Vol.2, Wiley Inter-science, New York,1963.
[20]J. P. Cox and R. T. Giuli, Principles of Stellar Structure, Vol. Ⅰ, pp.103-5, Gordon and Breach, New York,1968.
[21]C. M. Dafermos, Developement of singularities in the motion of materials with fading memory, Arch. Rational Mech. Anal.,91(1986),193-205.
[22]X. Ding, G.-Q. Chen, P. Lou, Convergence of the fraction step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Commun. Math. Phys., 121(1989),63-84.
[23]X. Ding, G.-Q. Chen and P. Lou, Convergence of the Lax-Friedrichs scheme for isen-tropic gas dynamics, Ⅰ,Ⅱ, Acta Math. Sci.,5(1985),415-432,433-472.
[24]R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal.,82(1983),27-70.
[25]R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Com-mun. Math. Phys.,91(1983),1-30.
[26]R. Diperna and P. L. Lions, On the Cauchy problem for the Boltzmann equation:Global existence and weak stability, Ann. Maths.,130(1989),321-366.
[27]B. Ducomet, A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas, Math. Mech. Appl. Sci.,22(1999),1323-1349.
[28]B. Ducomet, Hydrodynamics models of gaseous stars, Reviews of Math. Phys.,8(1996), 957-1000.
[29]B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics:on the interation between matter and radiation in the evlution of gaseous stars, Comm. Math. Phys., 266(2006),595-629.
[30]B. Ducomet and A. Zlotnik, Lyapunov functional methods for ID radiative and ractive viscous gas dynamics, Arch. Rational Mech. Anal.,177(2005),185-229.
[31]L. C. Evans, Partial Differential Equations, Gradudte Studies in Math. Vol.19,1998.
[32]R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics. Vol. Ⅲ, pp.4-7, Addison-Wesley, Reading, Massachusetts,1966.
[33]K. O. Friedrichs, Nonlinear hyperbolic differential equations for two independent vari-ables, Amer. J. Math.,70(1948),555-589.
[34]J. Glimm, Solutions in the large for nonlinear hyperbolic system of equations, Comm. Pure Appl. Math.,18(1965),95-105.
[35]S. Godunov, A difference method for numerical calculation of discontinuous solution-sof the equations of hydrodynamics, Mat. Sb.,47(89)(1959),271-360.
[36]M. Grassin, Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math.J.,47(1998),1397-1432.
[37]Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53(4)(2004),1081-1094.
[38]K. Hamdache, Existence in the large and asymptotic behavior for the Boltzmann equa-tion, Japan J. Appl. Math.,2(1985),1-15.
[39]L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a sys-tem of hyperbolic conservation laws with damping, Commun. Math. Phys.143(1992), 599-605.
[40]R. Illner and M.Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum, Commun. Math. Phys.,105(1986),189-203:Erratum and improved result, Commun. Math. Phys.,121(1989),143-146.
[41]K. Ito, BV-solutions of a hyperbolic-elliptic system for a radiation gas. Preprint, Hokkaide University,1997.
[42]S. Jiang and X. H. Zhong, Local existence and finite-time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid. Mech.,9(2007),543-564.
[43]F.John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math.,27(1974),377-405.
[44]S. Kaniel and M. Shinbrot, The Boltzmann equation. I. Uniqueness and local existence. Comm. Math. Phys.,59 (1978),65-84.
[45]T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. RationalMech. Anal,58(1975),181-205.
[46]S. Kawashima, System of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, thesis, Kyoto Univ.,1983.
[47]S. Kawashima, Y. Nikkuni and S. Nishibata, The Initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics. Analysis of sys-tems of conservation laws (Aachen,1997),87-127, Chapman & Hall/CRC, Monogr. Surv. Pure Appl. Math.,99, Chapman & Hall/CRC, Boca Raton, FL,1999.
[48]S. Kawashima and S. Nishibata, Cauchy problem for a model system of radiaing gas: weak solution with a jump and classical solutions, Math. Model Mech. Appl. Sci., 9(1999),69-91.
[49]S. Kawashima and S. Nishibata, Shock wave for a model system of the radiation gas, SIAM. J. Math. Anal,30(1)(1999),95-117.
[50]S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J Math,30(2004),211-250.
[51]E. H. Kennard, Knetic Theory of Gases, McGraw-Hill, New York,1938.
[52]S. Klainerman and Majda. A, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Appl. Math.,33(1980),241-263.
[53]L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory,2nd edn, pp.164,241, Addison-Wesley, Reading, Massachusetts,1965.
[54]P. D. Lax, Developement of singularities in solutions of nonlinear hyperbolic partial differential equations,J. Math. Phys.,5(1964),611-613.
[55]P. D. Lax, Hyperbolic Systems of Conservation Laws, Part Ⅱ, Comm Pure Appl Math, 10(1957),537-556.
[56]P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia,1972.
[57]P. D. Lax, Nonlinear hyperbolic equations, Comm. Pure Appl. Math.,6(1953),231-258.
[58]P. D. Lax, The formation and decay of shock waves, Amer. Math. Monthly,79(1972), 227-241.
[59]P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical com-putation, Comm, Pure Appl. Math.,7(1954),159-193.
[60]T.-T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Wiley, Chich-ester 1994.
[61]T.-T. Li and W.-C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke Univ. Press, Durham 1985.
[62]L.W. Lin, A note on developement of singularities of solutions of nonlinear hyperbolic partial differential equations, Advances in Nonlinear Partial Differertial Equations and Related Areas (Beijing 1997), G.-Q. Chen et al., eds, World Sci. Publishing, River Edge, NJ (1998),241-249.
[63]L.W. Lin, On the Vacuum state for the equations of isentropic gas dynamics,J. Math. Anal. Appl.,121(1987),406-425.
[64]P. L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of in-compressible fluid mechanics,I. Arch. Rational Mech. Anal,121(2001),173-193.
[65]P. L. Lions, B. Perthame and E. Souganisdis, Existence and stability of entropy solu-tions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math.,49(1996),599-638.
[66]P. L. Lions, B. Perthame, E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Commun. Math. Phys.,163(1994),415-431.
[67]T.-P. Liu The development of singularities in the nonlinear wave for quasi-linear hyper-bolic partial differential equations, J. Differ. Equations,33(1979),92-111.
[68]T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math.,57(12)(2004), 1543-1608.
[69]T. P. Liu and Y. Zeng, Large time behavior of solutions general quasi-linear hyperbolic-parabolic systems of conservation laws, A. M. S. memoirs,599(1997).
[70]L. Loeb, Kinetic Theory of Gases, Dover, New York,1961.
[71]A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences 53, Springer-Verlag:New York, 1984.
[72]T. Makino, S. Takeno, Initial boundary value problem for the spherically symmetric motion of isentropic gas, Japan J. Indust. Appl. Math.,11(1994),171-183.
[73]A. Matsumura and K. Nishihara, Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect, Quart. Appl. Math.,58(2000),69-83.
[74]A. Matsumura and K. Nishihara, On the stability of traveling wave solutions of a onedimensional model system for compressible viscous gas, Japan J. Appl. Math., 2(1)(1985),17-25.
[75]J. C. Maxwell, On the dynamical theory of gas, Philosophical Transactions of the Royal Society of London,157(1867),49-88.
[76]G. Metivier, Small Viscosity and Boundary Layer Methods-Theory, Stability Analysis, and Applications. Birkhauser, Boston,2003.
[77]D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics, New York: Oxford University Press,1984.
[78]G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press,1973.
[79]R. Racke, Lectures on Nonlinear Evolution Equations, Vieweg, Braun-schweig/wiesbaden,1992.
[80]M. A. Rammaha, Formation of singularities in compressible fluids in two-space dimen-sions, Proceedings of the American Mathematical Society,107(3)(1989),705-714.
[81]C. Rohde and W. A. Yong, The nonrelativistic limit in radiation hydrodynamics:I. weak entropy solutions for a model problem, J of Diff Eqs,234(2007),91-109.
[82]P. Secchi, On the motion of gaseous stars in the presence of radiation, Commu. PDE., 15(1990),185-204.
[83]T. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math.Phys.,101(1985),475-485.
[84]T. Sideris, Formation of singularities of solutions to nonlinear hyperbolic equations, Arch. Ration. Mech. Anal.,86(1984),369-381.
[85]T. Sideris, Global behavior of solutions to nonlinear wave equations in three space dimensions, Commun. Partial Differ. Equations,8(12)(1983),1291-1323.
[86]T. C. Sideris, B. Thomases, D. Wang, Long time behavior of solutions to the 3D com-pressible Euler equations with damping, Commu. PDE,28(3)(2003),795-816.
[87]J. Smoller, Shock Waves and Reaction-diffusion Equations, Springer, New York,1994.
[88]E. Tadmor, Approximate solutions of nonlinear conservation laws and related equa-tions, Recent Advances in Partial Differential Equations (Venice,1996), Proc. Sympos. Appl. Math., Vol.54, Amer. Math. Soc., Providence, RI (1998).
[89]L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics,665, Springer, Berlin (1977),228-241.
[90]L. Tartar, The compensated compactness method applied to system of conservation laws, Systems of Nonlinear P. D. E., J. M. Ball, ed., NATO Series, C. Reidel Publishing Co. (1983),263-285.
[91]S. Ukai, Solutions of the Boltzmann equation, Pattern and Waves Qualitave Analysis of Nonlinear Differential Equations (eds. M.Mimura and T.Nishida), Studies of Math-ematics and Its Applications 18, pp37-96, Kinokuniya-North-Holland, Tokyo,1986.
[92]S. Ukai and K. Asano, Stationary Solutions of the Boltzmann equation for a gas flow past an obstacle, Ⅰ. Existence, Arch. Rational Anal. Mech.,84(1983),248-291.
[93]S. Ukai and K. Asano, Stationary Solutions of the Boltzmann equation for a gas flow past an obstacle, Ⅱ. Stability, Publ. Res. Inst. Math. Sci., Kyoto Univ.,22(6)(1986), 1035-1062.
[94]S. Ukai, and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes, 2003.
[95]S. Ukai, T. Yang, and S.-H. Yu, Nonlinear stability of boundary layers of the Boltzmann equation:I, The case M∞<-1, Commun. Math. Phys.,244(2004),99-109.
[96]M. Umehara and A. Tani, Global solution to the one-dimensional equations for a self-gravitating viscoue radiative and ractive gas, J. Diff. Eqns.,234(2007),439-463.
[97]D. Wang, Global solutions and relaxation limits of Euler-Poisson equations, Z. angew. Math. Phys.,52(2001),620-630.
[98]D. Wang, Global solutions and stability for self-graritating isentropic gases, J. Math. Anal. Appl.,229(1999),530-542.
[99]D. Wang, Global solutions to the Euler-poisson equations of two-carrier types in one dimension, Z. angew. Math. Phys.,48(1997),680-693.
[100]YG. Wang, Nonlinear geometric optics for shock waves.Part 2:System case, Z. Anal. Anw.,16(1997),857-918.
[101]D. Wang and G.-Q. Chen, Formation of singularities in compressible Euler-Poisson fluids with heat diffusion and damping relaxation, J. Diff. Equations,144(1998),44-65.
[102]W. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, Journal of Diff. Equations,171(2001),410-450.
[103]Z. P. Xin, Analysis of some singular solutions in fluid dynamics, Proceedings of the International Congress of Mathematicians, Vol. Ⅲ (Beijing,2002),851-860. Higher Ed. Press, Beijing,2002.
[104]S.-H. Yu, Hydrodynamical limits with shock waves of the Boltzmann equation, Com-mun. Pur. Appl. Math.,58(2005),409-443.
[105]Y. B. Zeldovich and YP. Raizer, Physics of Shock Wave and High-Temperature Hy-drodynamics Phenomenon, Academic Press,1966.
[106]B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hy-drodynamic model for semiconductor devices, Commun. Math. Phys.,157(1993),1-22.
[2]R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal.,152(4)(2000),327-355.
[3]S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Birkhauser, Boston, (1972).
[4]C. Buet and B. Despres, Asymptotic analysis of fluid models for coupling of radiation and hydronamics, J. of Quantative Spectroscopy and Radiative Transfer,85(2004),385-418.
[5]C. Bardos, F. Golse and D. Levermore, Fluid dynamical limits of kinetic equations Formal derivation,J. Statistical.Phys.,63(1991),323-344;Ⅱ. Convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math.,46(1993),667-753.
[6]C. Bardos, F. Golse and D. Levermore, The acoustic limit of the Boltzmann equation, Arch. Raional Mech. Anal.,153(2000),177-204.
[7]P.L. Bhatnagar, E.P. Gross and M. Krook, A model for collision processes in gases.Ⅰ. small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94(1954),511-525.
[8]L. Boltzmann, Lectures on Gas Theory (Transl. by S. G. Brush), Dover Publ. Inc. New York,1964.
[9]A. Bressan, Hyperbolic Systems of Conservation Laws:The One Dimensional Cauchy Problem, Oxford Univ. Press, Oxford,2000.
[10]J. I. Castor, Radiation Hydrodynamics, Cambridge University Press,2004.
[11]G.-Q. Chen, Convergence of Lax-Friedrichs scheme for isentropic gas dynamics. Ⅲ, Acta Math. Sci.,59(1986),75-120.
[12]G.-Q. Chen, Entropy, Compactness, and Conservation Laws, Lecture Notes, North-western University,2001.
[13]G.-Q. Chen and T. H. Li, Global entropy solution in L∞ to the Euler equations and Euler-Poisson equatins for isothermal fluid with spherical symmetry, Methods and Ap-plications of Analysis.,10(2)(2003),215-244.
[14]G.-Q. Chen and D. Wang, Convergence of shock capturing scheme for the compressible Euler-Poisson equations, Commun. Math. Phys.,179(1996),333-364.
[15]G.-Q. Chen and D. Wang, The Cauchy problem for the Euler equations for compressible fluids, Handbook of Mathematical Fluid Dynamics, Amsterdam:North-Holland,2002, 421-543.
[16]G.-Q. Chen, Y. G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch Rational Mech Anal,179(3)(2008), 369-408.
[17]S. X. Chen, Stability of a Mach configuration, Comm Pure Appl Math,59(2006),1-35.
[18]R. Courant and K. O.Friedrichs, Supersonic Flow and Shock Wave, Springer-Verlag, Berlin, Heidelberg, New York,1976.
[19]R. Courant and D. Hilbert, Methods of Mathematical physics. Vol.2, Wiley Inter-science, New York,1963.
[20]J. P. Cox and R. T. Giuli, Principles of Stellar Structure, Vol. Ⅰ, pp.103-5, Gordon and Breach, New York,1968.
[21]C. M. Dafermos, Developement of singularities in the motion of materials with fading memory, Arch. Rational Mech. Anal.,91(1986),193-205.
[22]X. Ding, G.-Q. Chen, P. Lou, Convergence of the fraction step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Commun. Math. Phys., 121(1989),63-84.
[23]X. Ding, G.-Q. Chen and P. Lou, Convergence of the Lax-Friedrichs scheme for isen-tropic gas dynamics, Ⅰ,Ⅱ, Acta Math. Sci.,5(1985),415-432,433-472.
[24]R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal.,82(1983),27-70.
[25]R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Com-mun. Math. Phys.,91(1983),1-30.
[26]R. Diperna and P. L. Lions, On the Cauchy problem for the Boltzmann equation:Global existence and weak stability, Ann. Maths.,130(1989),321-366.
[27]B. Ducomet, A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas, Math. Mech. Appl. Sci.,22(1999),1323-1349.
[28]B. Ducomet, Hydrodynamics models of gaseous stars, Reviews of Math. Phys.,8(1996), 957-1000.
[29]B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics:on the interation between matter and radiation in the evlution of gaseous stars, Comm. Math. Phys., 266(2006),595-629.
[30]B. Ducomet and A. Zlotnik, Lyapunov functional methods for ID radiative and ractive viscous gas dynamics, Arch. Rational Mech. Anal.,177(2005),185-229.
[31]L. C. Evans, Partial Differential Equations, Gradudte Studies in Math. Vol.19,1998.
[32]R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics. Vol. Ⅲ, pp.4-7, Addison-Wesley, Reading, Massachusetts,1966.
[33]K. O. Friedrichs, Nonlinear hyperbolic differential equations for two independent vari-ables, Amer. J. Math.,70(1948),555-589.
[34]J. Glimm, Solutions in the large for nonlinear hyperbolic system of equations, Comm. Pure Appl. Math.,18(1965),95-105.
[35]S. Godunov, A difference method for numerical calculation of discontinuous solution-sof the equations of hydrodynamics, Mat. Sb.,47(89)(1959),271-360.
[36]M. Grassin, Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math.J.,47(1998),1397-1432.
[37]Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53(4)(2004),1081-1094.
[38]K. Hamdache, Existence in the large and asymptotic behavior for the Boltzmann equa-tion, Japan J. Appl. Math.,2(1985),1-15.
[39]L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a sys-tem of hyperbolic conservation laws with damping, Commun. Math. Phys.143(1992), 599-605.
[40]R. Illner and M.Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum, Commun. Math. Phys.,105(1986),189-203:Erratum and improved result, Commun. Math. Phys.,121(1989),143-146.
[41]K. Ito, BV-solutions of a hyperbolic-elliptic system for a radiation gas. Preprint, Hokkaide University,1997.
[42]S. Jiang and X. H. Zhong, Local existence and finite-time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid. Mech.,9(2007),543-564.
[43]F.John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math.,27(1974),377-405.
[44]S. Kaniel and M. Shinbrot, The Boltzmann equation. I. Uniqueness and local existence. Comm. Math. Phys.,59 (1978),65-84.
[45]T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. RationalMech. Anal,58(1975),181-205.
[46]S. Kawashima, System of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, thesis, Kyoto Univ.,1983.
[47]S. Kawashima, Y. Nikkuni and S. Nishibata, The Initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics. Analysis of sys-tems of conservation laws (Aachen,1997),87-127, Chapman & Hall/CRC, Monogr. Surv. Pure Appl. Math.,99, Chapman & Hall/CRC, Boca Raton, FL,1999.
[48]S. Kawashima and S. Nishibata, Cauchy problem for a model system of radiaing gas: weak solution with a jump and classical solutions, Math. Model Mech. Appl. Sci., 9(1999),69-91.
[49]S. Kawashima and S. Nishibata, Shock wave for a model system of the radiation gas, SIAM. J. Math. Anal,30(1)(1999),95-117.
[50]S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J Math,30(2004),211-250.
[51]E. H. Kennard, Knetic Theory of Gases, McGraw-Hill, New York,1938.
[52]S. Klainerman and Majda. A, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Appl. Math.,33(1980),241-263.
[53]L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory,2nd edn, pp.164,241, Addison-Wesley, Reading, Massachusetts,1965.
[54]P. D. Lax, Developement of singularities in solutions of nonlinear hyperbolic partial differential equations,J. Math. Phys.,5(1964),611-613.
[55]P. D. Lax, Hyperbolic Systems of Conservation Laws, Part Ⅱ, Comm Pure Appl Math, 10(1957),537-556.
[56]P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia,1972.
[57]P. D. Lax, Nonlinear hyperbolic equations, Comm. Pure Appl. Math.,6(1953),231-258.
[58]P. D. Lax, The formation and decay of shock waves, Amer. Math. Monthly,79(1972), 227-241.
[59]P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical com-putation, Comm, Pure Appl. Math.,7(1954),159-193.
[60]T.-T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Wiley, Chich-ester 1994.
[61]T.-T. Li and W.-C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke Univ. Press, Durham 1985.
[62]L.W. Lin, A note on developement of singularities of solutions of nonlinear hyperbolic partial differential equations, Advances in Nonlinear Partial Differertial Equations and Related Areas (Beijing 1997), G.-Q. Chen et al., eds, World Sci. Publishing, River Edge, NJ (1998),241-249.
[63]L.W. Lin, On the Vacuum state for the equations of isentropic gas dynamics,J. Math. Anal. Appl.,121(1987),406-425.
[64]P. L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of in-compressible fluid mechanics,I. Arch. Rational Mech. Anal,121(2001),173-193.
[65]P. L. Lions, B. Perthame and E. Souganisdis, Existence and stability of entropy solu-tions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math.,49(1996),599-638.
[66]P. L. Lions, B. Perthame, E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Commun. Math. Phys.,163(1994),415-431.
[67]T.-P. Liu The development of singularities in the nonlinear wave for quasi-linear hyper-bolic partial differential equations, J. Differ. Equations,33(1979),92-111.
[68]T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math.,57(12)(2004), 1543-1608.
[69]T. P. Liu and Y. Zeng, Large time behavior of solutions general quasi-linear hyperbolic-parabolic systems of conservation laws, A. M. S. memoirs,599(1997).
[70]L. Loeb, Kinetic Theory of Gases, Dover, New York,1961.
[71]A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences 53, Springer-Verlag:New York, 1984.
[72]T. Makino, S. Takeno, Initial boundary value problem for the spherically symmetric motion of isentropic gas, Japan J. Indust. Appl. Math.,11(1994),171-183.
[73]A. Matsumura and K. Nishihara, Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect, Quart. Appl. Math.,58(2000),69-83.
[74]A. Matsumura and K. Nishihara, On the stability of traveling wave solutions of a onedimensional model system for compressible viscous gas, Japan J. Appl. Math., 2(1)(1985),17-25.
[75]J. C. Maxwell, On the dynamical theory of gas, Philosophical Transactions of the Royal Society of London,157(1867),49-88.
[76]G. Metivier, Small Viscosity and Boundary Layer Methods-Theory, Stability Analysis, and Applications. Birkhauser, Boston,2003.
[77]D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics, New York: Oxford University Press,1984.
[78]G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press,1973.
[79]R. Racke, Lectures on Nonlinear Evolution Equations, Vieweg, Braun-schweig/wiesbaden,1992.
[80]M. A. Rammaha, Formation of singularities in compressible fluids in two-space dimen-sions, Proceedings of the American Mathematical Society,107(3)(1989),705-714.
[81]C. Rohde and W. A. Yong, The nonrelativistic limit in radiation hydrodynamics:I. weak entropy solutions for a model problem, J of Diff Eqs,234(2007),91-109.
[82]P. Secchi, On the motion of gaseous stars in the presence of radiation, Commu. PDE., 15(1990),185-204.
[83]T. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math.Phys.,101(1985),475-485.
[84]T. Sideris, Formation of singularities of solutions to nonlinear hyperbolic equations, Arch. Ration. Mech. Anal.,86(1984),369-381.
[85]T. Sideris, Global behavior of solutions to nonlinear wave equations in three space dimensions, Commun. Partial Differ. Equations,8(12)(1983),1291-1323.
[86]T. C. Sideris, B. Thomases, D. Wang, Long time behavior of solutions to the 3D com-pressible Euler equations with damping, Commu. PDE,28(3)(2003),795-816.
[87]J. Smoller, Shock Waves and Reaction-diffusion Equations, Springer, New York,1994.
[88]E. Tadmor, Approximate solutions of nonlinear conservation laws and related equa-tions, Recent Advances in Partial Differential Equations (Venice,1996), Proc. Sympos. Appl. Math., Vol.54, Amer. Math. Soc., Providence, RI (1998).
[89]L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics,665, Springer, Berlin (1977),228-241.
[90]L. Tartar, The compensated compactness method applied to system of conservation laws, Systems of Nonlinear P. D. E., J. M. Ball, ed., NATO Series, C. Reidel Publishing Co. (1983),263-285.
[91]S. Ukai, Solutions of the Boltzmann equation, Pattern and Waves Qualitave Analysis of Nonlinear Differential Equations (eds. M.Mimura and T.Nishida), Studies of Math-ematics and Its Applications 18, pp37-96, Kinokuniya-North-Holland, Tokyo,1986.
[92]S. Ukai and K. Asano, Stationary Solutions of the Boltzmann equation for a gas flow past an obstacle, Ⅰ. Existence, Arch. Rational Anal. Mech.,84(1983),248-291.
[93]S. Ukai and K. Asano, Stationary Solutions of the Boltzmann equation for a gas flow past an obstacle, Ⅱ. Stability, Publ. Res. Inst. Math. Sci., Kyoto Univ.,22(6)(1986), 1035-1062.
[94]S. Ukai, and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes, 2003.
[95]S. Ukai, T. Yang, and S.-H. Yu, Nonlinear stability of boundary layers of the Boltzmann equation:I, The case M∞<-1, Commun. Math. Phys.,244(2004),99-109.
[96]M. Umehara and A. Tani, Global solution to the one-dimensional equations for a self-gravitating viscoue radiative and ractive gas, J. Diff. Eqns.,234(2007),439-463.
[97]D. Wang, Global solutions and relaxation limits of Euler-Poisson equations, Z. angew. Math. Phys.,52(2001),620-630.
[98]D. Wang, Global solutions and stability for self-graritating isentropic gases, J. Math. Anal. Appl.,229(1999),530-542.
[99]D. Wang, Global solutions to the Euler-poisson equations of two-carrier types in one dimension, Z. angew. Math. Phys.,48(1997),680-693.
[100]YG. Wang, Nonlinear geometric optics for shock waves.Part 2:System case, Z. Anal. Anw.,16(1997),857-918.
[101]D. Wang and G.-Q. Chen, Formation of singularities in compressible Euler-Poisson fluids with heat diffusion and damping relaxation, J. Diff. Equations,144(1998),44-65.
[102]W. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, Journal of Diff. Equations,171(2001),410-450.
[103]Z. P. Xin, Analysis of some singular solutions in fluid dynamics, Proceedings of the International Congress of Mathematicians, Vol. Ⅲ (Beijing,2002),851-860. Higher Ed. Press, Beijing,2002.
[104]S.-H. Yu, Hydrodynamical limits with shock waves of the Boltzmann equation, Com-mun. Pur. Appl. Math.,58(2005),409-443.
[105]Y. B. Zeldovich and YP. Raizer, Physics of Shock Wave and High-Temperature Hy-drodynamics Phenomenon, Academic Press,1966.
[106]B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hy-drodynamic model for semiconductor devices, Commun. Math. Phys.,157(1993),1-22.