# American Institute of Mathematical Sciences

May  2012, 17(3): 1075-1100. doi: 10.3934/dcdsb.2012.17.1075

## Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model

 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received  August 2011 Revised  September 2011 Published  January 2012

In this paper we consider the large-time behavior of solutions for the Cauchy problem to a compressible radiating gas model, where the far field states are prescribed. This radiating gas model is represented by the one-dimensional system of gas dynamics coupled with an elliptic equation for radiation flux. When the corresponding Riemann problem for the compressible Euler system admits a solution consisting of a contact wave and two rarefaction waves, it is proved that for such a radiating gas model, the combination of viscous contact wave with rarefaction waves is asymptotically stable provided that the strength of combination wave is suitably small. This result is proved by a domain decomposition technique and elementary energy methods.
Citation: 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
##### References:
 [1] D. G. Aronson, The porous media equation,, in, 1224 (1986), 1.   Google Scholar [2] 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 [3] K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas,, Quart. J. Mech. Appl. Math., 24 (1971), 155.  doi: 10.1093/qjmam/24.2.155.  Google Scholar [4] F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations,, Arch. Ration. Mech. Anal., 179 (2006), 55.  doi: 10.1007/s00205-005-0380-7.  Google Scholar [5] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions,, Adv. Math., 219 (2008), 1246.  doi: 10.1016/j.aim.2008.06.014.  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. Rational Mech. Anal., 197 (2010), 89.  doi: 10.1007/s00205-009-0267-0.  Google Scholar [7] S. Jiang, G. Ni and W. Sun, Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids,, SIAM J. Math. Anal., 38 (2006), 368.  doi: 10.1137/050626478.  Google Scholar [8] S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics,, in, 99 (1999), 87.   Google Scholar [9] S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gas,, SIAM J. Math. Anal., 30 (1999), 95.  doi: 10.1137/S0036141097322169.  Google Scholar [10] 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 [11] S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas,, Kyushu J. Math., 58 (2004), 211.  doi: 10.2206/kyushujm.58.211.  Google Scholar [12] 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 [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.   Google Scholar [15] C. J. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics,, Comm. Math. Sci., 9 (2011), 207.   Google Scholar [16] T.-P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations,, Comm. Math. Phys., 118 (1988), 451.  doi: 10.1007/BF01466726.  Google Scholar [17] T.-P. Liu and Z. P. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws,, Asian J. Math., 1 (1997), 34.   Google Scholar [18] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas,, Japan J. Appl. Math., 3 (1986), 1.  doi: 10.1007/BF03167088.  Google Scholar [19] 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 [20] K. Nishihara, T. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations,, SIAM J. Math. Ana., 35 (2004), 1561.  doi: 10.1137/S003614100342735X.  Google Scholar [21] C. Rohde and F. Xie, Decay rates to viscous contact wave for a 1D compressible radiation hydrodynamics model,, submitted, (2011).   Google Scholar [22] L. Z. Ruan and C. J. Zhu, Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas,, J. Differential Equations, 249 (2010), 2076.  doi: 10.1016/j.jde.2010.07.029.  Google Scholar [23] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, 258 (1994).   Google Scholar [24] C. J. van Duyn and L. A. Peletier, A class of similarity solutions of the nonlinear diffusion equation,, Nonlinear Anal., 1 (): 223.   Google Scholar [25] W. Vincenti and C. Kruger, "Introduction to Physical Gas Dynamics,", Wiley, (1965).   Google Scholar [26] J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system,, Jour. Diff. Equations, 251 (2011), 1030.  doi: 10.1016/j.jde.2011.03.011.  Google Scholar [27] 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 [28] J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the one-dimensional compressible viscous gas with radiation,, Nonlinear Analysis, 74 (2011), 4138.  doi: 10.1016/j.na.2011.03.047.  Google Scholar [29] Z. P. Xin, On nonlinear stability of contact discontinuities,, in, (1996), 249.   Google Scholar

show all references

##### References:
 [1] D. G. Aronson, The porous media equation,, in, 1224 (1986), 1.   Google Scholar [2] 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 [3] K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas,, Quart. J. Mech. Appl. Math., 24 (1971), 155.  doi: 10.1093/qjmam/24.2.155.  Google Scholar [4] F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations,, Arch. Ration. Mech. Anal., 179 (2006), 55.  doi: 10.1007/s00205-005-0380-7.  Google Scholar [5] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions,, Adv. Math., 219 (2008), 1246.  doi: 10.1016/j.aim.2008.06.014.  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. Rational Mech. Anal., 197 (2010), 89.  doi: 10.1007/s00205-009-0267-0.  Google Scholar [7] S. Jiang, G. Ni and W. Sun, Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids,, SIAM J. Math. Anal., 38 (2006), 368.  doi: 10.1137/050626478.  Google Scholar [8] S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics,, in, 99 (1999), 87.   Google Scholar [9] S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gas,, SIAM J. Math. Anal., 30 (1999), 95.  doi: 10.1137/S0036141097322169.  Google Scholar [10] 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 [11] S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas,, Kyushu J. Math., 58 (2004), 211.  doi: 10.2206/kyushujm.58.211.  Google Scholar [12] 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 [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.   Google Scholar [15] C. J. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics,, Comm. Math. Sci., 9 (2011), 207.   Google Scholar [16] T.-P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations,, Comm. Math. Phys., 118 (1988), 451.  doi: 10.1007/BF01466726.  Google Scholar [17] T.-P. Liu and Z. P. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws,, Asian J. Math., 1 (1997), 34.   Google Scholar [18] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas,, Japan J. Appl. Math., 3 (1986), 1.  doi: 10.1007/BF03167088.  Google Scholar [19] 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 [20] K. Nishihara, T. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations,, SIAM J. Math. Ana., 35 (2004), 1561.  doi: 10.1137/S003614100342735X.  Google Scholar [21] C. Rohde and F. Xie, Decay rates to viscous contact wave for a 1D compressible radiation hydrodynamics model,, submitted, (2011).   Google Scholar [22] L. Z. Ruan and C. J. Zhu, Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas,, J. Differential Equations, 249 (2010), 2076.  doi: 10.1016/j.jde.2010.07.029.  Google Scholar [23] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, 258 (1994).   Google Scholar [24] C. J. van Duyn and L. A. Peletier, A class of similarity solutions of the nonlinear diffusion equation,, Nonlinear Anal., 1 (): 223.   Google Scholar [25] W. Vincenti and C. Kruger, "Introduction to Physical Gas Dynamics,", Wiley, (1965).   Google Scholar [26] J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system,, Jour. Diff. Equations, 251 (2011), 1030.  doi: 10.1016/j.jde.2011.03.011.  Google Scholar [27] 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 [28] J. Wang and F. Xie, Asymptotic stability of viscous contact wave for the one-dimensional compressible viscous gas with radiation,, Nonlinear Analysis, 74 (2011), 4138.  doi: 10.1016/j.na.2011.03.047.  Google Scholar [29] Z. P. Xin, On nonlinear stability of contact discontinuities,, in, (1996), 249.   Google Scholar
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