# 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] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [2] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [3] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434 [4] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [5] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [6] Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270 [7] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [8] Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108 [9] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [10] Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215 [11] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [12] Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466 [13] Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $L^2$-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 [14] Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 [15] George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 [16] Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 [17] Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 [18] Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A socp relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104 [19] Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 [20] Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

2019 Impact Factor: 1.105