# American Institute of Mathematical Sciences

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

 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.
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 & 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., (1986).   Google Scholar [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.  doi: 10.1007/s00030-006-4023-y.  Google Scholar [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.  doi: 10.1142/S0218202508002760.  Google Scholar [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.  doi: 10.1016/j.jde.2008.02.023.  Google Scholar [5] K. Hamer, Nonlinear effects on the propagation of sounds waves in a radiating gas,, Quarter J. Mech. Appl. Math., 24 (1971), 155.  doi: 10.1093/qjmam/24.2.155.  Google Scholar [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.  doi: 10.1007/s00205-009-0267-0.  Google Scholar [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.  doi: 10.1007/s00205-002-0234-5.  Google Scholar [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.  doi: 10.3934/krm.2010.3.685.  Google Scholar [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, (1997), 87.   Google Scholar [10] S. Kawashima and S. Nishibata, A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics,, Indiana Univ. Math. J., 101 (1985), 97.   Google Scholar [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.  doi: 10.1007/s00205-003-0273-6.  Google Scholar [12] C. J. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics,, Commun. Math. Sci., 9 (2011), 207.   Google Scholar [13] C. J. Lin, J. F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases,, Phys. D, 218 (2006), 83.  doi: 10.1016/j.physd.2006.04.012.  Google Scholar [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.  doi: 10.1016/j.crma.2007.10.029.  Google Scholar [15] C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas,, J. Differential Equations, 190 (2003), 439.  doi: 10.1016/S0022-0396(02)00158-4.  Google Scholar [16] C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems,, Indiana Univ. Math. J., 56 (2007), 2601.  doi: 10.1512/iumj.2007.56.3043.  Google Scholar [17] T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems,, Phys. D, 239 (2010), 428.  doi: 10.1016/j.physd.2010.01.011.  Google Scholar [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)., (2012).   Google Scholar [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.  doi: 10.1016/j.jde.2006.11.010.  Google Scholar [20] J. Smoller, "Shock Waves and Reaction-diffusion Equations,'', Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar [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.  Google Scholar [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.  doi: 10.1137/100792792.  Google Scholar [23] J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system,, J. Differential Equations, 251 (2011), 1030.  doi: 10.1016/j.jde.2011.03.011.  Google Scholar [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.  doi: 10.3934/dcdsb.2012.17.1075.  Google Scholar [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.  doi: 10.1002/cpa.3160460502.  Google Scholar

show all references

##### References:
 [1] D. G. Aronson, The porous media equations, in "Nonlinear Diffusion Problem,'', Lecture Notes in Math., (1986).   Google Scholar [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.  doi: 10.1007/s00030-006-4023-y.  Google Scholar [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.  doi: 10.1142/S0218202508002760.  Google Scholar [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.  doi: 10.1016/j.jde.2008.02.023.  Google Scholar [5] K. Hamer, Nonlinear effects on the propagation of sounds waves in a radiating gas,, Quarter J. Mech. Appl. Math., 24 (1971), 155.  doi: 10.1093/qjmam/24.2.155.  Google Scholar [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.  doi: 10.1007/s00205-009-0267-0.  Google Scholar [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.  doi: 10.1007/s00205-002-0234-5.  Google Scholar [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.  doi: 10.3934/krm.2010.3.685.  Google Scholar [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, (1997), 87.   Google Scholar [10] S. Kawashima and S. Nishibata, A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics,, Indiana Univ. Math. J., 101 (1985), 97.   Google Scholar [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.  doi: 10.1007/s00205-003-0273-6.  Google Scholar [12] C. J. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics,, Commun. Math. Sci., 9 (2011), 207.   Google Scholar [13] C. J. Lin, J. F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases,, Phys. D, 218 (2006), 83.  doi: 10.1016/j.physd.2006.04.012.  Google Scholar [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.  doi: 10.1016/j.crma.2007.10.029.  Google Scholar [15] C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas,, J. Differential Equations, 190 (2003), 439.  doi: 10.1016/S0022-0396(02)00158-4.  Google Scholar [16] C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems,, Indiana Univ. Math. J., 56 (2007), 2601.  doi: 10.1512/iumj.2007.56.3043.  Google Scholar [17] T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems,, Phys. D, 239 (2010), 428.  doi: 10.1016/j.physd.2010.01.011.  Google Scholar [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)., (2012).   Google Scholar [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.  doi: 10.1016/j.jde.2006.11.010.  Google Scholar [20] J. Smoller, "Shock Waves and Reaction-diffusion Equations,'', Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar [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.  Google Scholar [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.  doi: 10.1137/100792792.  Google Scholar [23] J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system,, J. Differential Equations, 251 (2011), 1030.  doi: 10.1016/j.jde.2011.03.011.  Google Scholar [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.  doi: 10.3934/dcdsb.2012.17.1075.  Google Scholar [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.  doi: 10.1002/cpa.3160460502.  Google Scholar
 [1] Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075 [2] Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907 [3] Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010 [4] Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic & Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685 [5] Haiyan Yin. The stability of contact discontinuity for compressible planar magnetohydrodynamics. Kinetic & Related Models, 2017, 10 (4) : 1235-1253. doi: 10.3934/krm.2017047 [6] Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271 [7] Eun Heui Kim, Charis Tsikkou. Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6257-6289. doi: 10.3934/dcds.2017271 [8] Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735 [9] Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227 [10] Benjamin Jourdain, Julien Reygner. Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4963-4996. doi: 10.3934/dcds.2016015 [11] Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031 [12] Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035 [13] Vladimir V. Chepyzhov, Anna Kostianko, Sergey Zelik. Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1115-1142. doi: 10.3934/dcdsb.2019009 [14] Josephus Hulshof, Pascal Noble. Travelling waves for a combustion model coupled with hyperbolic radiation moment models. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 73-90. doi: 10.3934/dcdsb.2008.10.73 [15] Huijiang Zhao, Yinchuan Zhao. Convergence to strong nonlinear rarefaction waves for global smooth solutions of $p-$system with relaxation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1243-1262. doi: 10.3934/dcds.2003.9.1243 [16] Teng Wang, Yi Wang. Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation. Kinetic & Related Models, 2019, 12 (3) : 637-679. doi: 10.3934/krm.2019025 [17] Claude-Michael Brauner, Josephus Hulshof, J.-F. Ripoll. Existence of travelling wave solutions in a combustion-radiation model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 193-208. doi: 10.3934/dcdsb.2001.1.193 [18] V. Styles. A note on the convergence in the limit of a long wave vortex density superconductivity model to the Bean model. Communications on Pure & Applied Analysis, 2002, 1 (4) : 485-494. doi: 10.3934/cpaa.2002.1.485 [19] T. Colin, D. Lannes. Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 83-100. doi: 10.3934/dcds.2004.11.83 [20] Weishi Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 871-884. doi: 10.3934/dcds.2004.10.871

2019 Impact Factor: 1.105