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Approximation of the trajectory attractor of the 3D MHD System
Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: superposition of rarefaction and contact waves
1. | Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart |
2. | College of Science, University of Shanghai for Science and Technology, Shanghai 200240, China |
3. | Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240 |
References:
[1] |
D. G. Aronson, The porous media equations, in "Nonlinear Diffusion Problem,'' Lecture Notes in Math., Vol. 1224, (A. Fasano and M. Primicerio eds.) Springer-Verlag, Berlin, (1986). |
[2] |
M. Di Francesco, Initial value problem and relaxation limits of the Hamer model for radiating gases in several space variables, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 531-562.
doi: 10.1007/s00030-006-4023-y. |
[3] |
W. L. Gao and C. J. Zhu, Asymptotic decay toward the planar rarefaction waves for a model system of the radiating gas in two dimensions, Math. Models Methods Appl. Sci., 18 (2008), 511-541.
doi: 10.1142/S0218202508002760. |
[4] |
W. L. Gao, L. Z. Ruan and C. J. Zhu, Decay rates to the planar rarefaction waves for a model system of the radiating gas in $n$n dimensions, J. Differential Equations, 244 (2008), 2614-2640.
doi: 10.1016/j.jde.2008.02.023. |
[5] |
K. Hamer, Nonlinear effects on the propagation of sounds waves in a radiating gas, Quarter J. Mech. Appl. Math., 24 (1971), 155-168.
doi: 10.1093/qjmam/24.2.155. |
[6] |
F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.
doi: 10.1007/s00205-009-0267-0. |
[7] |
F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376.
doi: 10.1007/s00205-002-0234-5. |
[8] |
F. M. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuiy, Kinetic and Related Models, 3 (2010), 685-728.
doi: 10.3934/krm.2010.3.685. |
[9] |
S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics, Analysis of Systems of Conservation Laws, (Aachen, 1997), 87-127. |
[10] |
S. Kawashima and S. Nishibata, A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics, Indiana Univ. Math. J., 101 (1985), 97-127. |
[11] |
S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems, Arch. Ration. Mech. Anal., 170 (2003), 297-329.
doi: 10.1007/s00205-003-0273-6. |
[12] |
C. J. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics, Commun. Math. Sci., 9 (2011), 207-223. |
[13] |
C. J. Lin, J. F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases, Phys. D, 218 (2006), 83-94.
doi: 10.1016/j.physd.2006.04.012. |
[14] |
C. J. Lin, J. F. Coulombel and T. Goudon, Asymptotic stability of shock profiles in radiative hydrodynamics, C. R. Math. Acad. Sci. Paris, 345 (2007), 625-628.
doi: 10.1016/j.crma.2007.10.029. |
[15] |
C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations, 190 (2003), 439-465.
doi: 10.1016/S0022-0396(02)00158-4. |
[16] |
C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems, Indiana Univ. Math. J., 56 (2007), 2601-2640.
doi: 10.1512/iumj.2007.56.3043. |
[17] |
T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Phys. D, 239 (2010), 428-453.
doi: 10.1016/j.physd.2010.01.011. |
[18] |
C. Rohde and F. Xie, Decay rates to viscous contact wave for a 1D compressible radiation hydrodynamics model, Math. Models Meth. Appl. Sci. DOI: 10.1142/S0218202512500522 (2012). |
[19] |
C. Rohde and W. A. Yong, The nonrelativistic limit in radiation hydrodynamics. I. Weak entropy solutions for a model problem, J. Differential Equations, 234 (2007), 91-109.
doi: 10.1016/j.jde.2006.11.010. |
[20] |
J. Smoller, "Shock Waves and Reaction-diffusion Equations,'' Springer-Verlag, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[21] |
C. J. van Duijn and L. A. Peletier, A class of similarity solutions of the nonlinear diffusion equation,, Nonlinear Anal., 1 (): 223.
doi: 10.1016/0362-546X(77)90032-3. |
[22] |
J. Wang and F. Xie, Singular limit to strong contact discontinuity for a 1D compressible radiation hydrodynamics model, SIAM J. Math. Anal., 43 (2011), 1189-1204.
doi: 10.1137/100792792. |
[23] |
J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system, J. Differential Equations, 251 (2011), 1030-1055.
doi: 10.1016/j.jde.2011.03.011. |
[24] |
F. Xie, Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model, Discrete and Continuous Dynam. Systems - B, 17 (2012), 1075-1100.
doi: 10.3934/dcdsb.2012.17.1075. |
[25] |
Z. P. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (1993), 621-665.
doi: 10.1002/cpa.3160460502. |
show all references
References:
[1] |
D. G. Aronson, The porous media equations, in "Nonlinear Diffusion Problem,'' Lecture Notes in Math., Vol. 1224, (A. Fasano and M. Primicerio eds.) Springer-Verlag, Berlin, (1986). |
[2] |
M. Di Francesco, Initial value problem and relaxation limits of the Hamer model for radiating gases in several space variables, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 531-562.
doi: 10.1007/s00030-006-4023-y. |
[3] |
W. L. Gao and C. J. Zhu, Asymptotic decay toward the planar rarefaction waves for a model system of the radiating gas in two dimensions, Math. Models Methods Appl. Sci., 18 (2008), 511-541.
doi: 10.1142/S0218202508002760. |
[4] |
W. L. Gao, L. Z. Ruan and C. J. Zhu, Decay rates to the planar rarefaction waves for a model system of the radiating gas in $n$n dimensions, J. Differential Equations, 244 (2008), 2614-2640.
doi: 10.1016/j.jde.2008.02.023. |
[5] |
K. Hamer, Nonlinear effects on the propagation of sounds waves in a radiating gas, Quarter J. Mech. Appl. Math., 24 (1971), 155-168.
doi: 10.1093/qjmam/24.2.155. |
[6] |
F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.
doi: 10.1007/s00205-009-0267-0. |
[7] |
F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376.
doi: 10.1007/s00205-002-0234-5. |
[8] |
F. M. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuiy, Kinetic and Related Models, 3 (2010), 685-728.
doi: 10.3934/krm.2010.3.685. |
[9] |
S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics, Analysis of Systems of Conservation Laws, (Aachen, 1997), 87-127. |
[10] |
S. Kawashima and S. Nishibata, A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics, Indiana Univ. Math. J., 101 (1985), 97-127. |
[11] |
S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems, Arch. Ration. Mech. Anal., 170 (2003), 297-329.
doi: 10.1007/s00205-003-0273-6. |
[12] |
C. J. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics, Commun. Math. Sci., 9 (2011), 207-223. |
[13] |
C. J. Lin, J. F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases, Phys. D, 218 (2006), 83-94.
doi: 10.1016/j.physd.2006.04.012. |
[14] |
C. J. Lin, J. F. Coulombel and T. Goudon, Asymptotic stability of shock profiles in radiative hydrodynamics, C. R. Math. Acad. Sci. Paris, 345 (2007), 625-628.
doi: 10.1016/j.crma.2007.10.029. |
[15] |
C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations, 190 (2003), 439-465.
doi: 10.1016/S0022-0396(02)00158-4. |
[16] |
C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems, Indiana Univ. Math. J., 56 (2007), 2601-2640.
doi: 10.1512/iumj.2007.56.3043. |
[17] |
T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Phys. D, 239 (2010), 428-453.
doi: 10.1016/j.physd.2010.01.011. |
[18] |
C. Rohde and F. Xie, Decay rates to viscous contact wave for a 1D compressible radiation hydrodynamics model, Math. Models Meth. Appl. Sci. DOI: 10.1142/S0218202512500522 (2012). |
[19] |
C. Rohde and W. A. Yong, The nonrelativistic limit in radiation hydrodynamics. I. Weak entropy solutions for a model problem, J. Differential Equations, 234 (2007), 91-109.
doi: 10.1016/j.jde.2006.11.010. |
[20] |
J. Smoller, "Shock Waves and Reaction-diffusion Equations,'' Springer-Verlag, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[21] |
C. J. van Duijn and L. A. Peletier, A class of similarity solutions of the nonlinear diffusion equation,, Nonlinear Anal., 1 (): 223.
doi: 10.1016/0362-546X(77)90032-3. |
[22] |
J. Wang and F. Xie, Singular limit to strong contact discontinuity for a 1D compressible radiation hydrodynamics model, SIAM J. Math. Anal., 43 (2011), 1189-1204.
doi: 10.1137/100792792. |
[23] |
J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system, J. Differential Equations, 251 (2011), 1030-1055.
doi: 10.1016/j.jde.2011.03.011. |
[24] |
F. Xie, Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model, Discrete and Continuous Dynam. Systems - B, 17 (2012), 1075-1100.
doi: 10.3934/dcdsb.2012.17.1075. |
[25] |
Z. P. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (1993), 621-665.
doi: 10.1002/cpa.3160460502. |
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