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September  2013, 12(5): 2145-2171. doi: 10.3934/cpaa.2013.12.2145

Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: superposition of rarefaction and contact waves

Christian Rohde 1, , Wenjun Wang 2, and  Feng Xie 3,

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

Received  May 2012 Revised  August 2012 Published  January 2013

In this paper we consider a hyperbolic-hyperbolic relaxation limit problem for a 1D compressible radiation hydrodynamics (RHD) system. The RHD system consists of the full Euler system coupled with an elliptic equation for the radiation flux. The singular relaxation limit process we consider corresponds to the physical problem of letting the Bouguer number become infinite. We prove for appropriate initial datum that the solution of the initial value problem for the RHD system converges for vanishing reciprocal Bouguer number to a weak solution of the limit system which is the Euler system. The initial data are chosen such that the limit solution is composed by a $1$-rarefaction wave, a contact discontinuity and a $3$-rarefaction wave. Moreover we give the convergence rate in terms of the physical parameter.
Keywords: convergence rate., Compressible radiation hydrodynamics, contact discontinuity wave, rarefaction wave, hyperbolic-hyperbolic relaxation limit.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 35L03, 35L6.
Citation: Christian Rohde, Wenjun Wang, Feng Xie. Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: superposition of rarefaction and contact waves. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2145-2171. doi: 10.3934/cpaa.2013.12.2145
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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