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 "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Math., 1224, Springer, Berlin, (1986), 1-46.  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), 511-541. 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), 155-168. doi: 10.1093/qjmam/24.2.155.  Google Scholar

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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), 55-77. 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), 1246-1297. 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), 89-116. 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), 368-384. 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 "Analysis of Systems of Conservation Laws" (Aachen, 1997), Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 99, Chapman & Hall/CRC, Boca Raton, FL, (1999), 87-127.  Google Scholar

[9]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117. 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), 297-329. 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), 211-250. 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), 2601-2640. 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), 83-94. 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), 625-628.  Google Scholar

[15]

C. J. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics, Comm. Math. Sci., 9 (2011), 207-223. 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), 451-465. 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), 34-84.  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), 1-13. 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), 428-453. 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), 1561-1597. 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), 2076-2110. doi: 10.1016/j.jde.2010.07.029.  Google Scholar

[23]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 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, New York, 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), 1030-1055. 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), 1189-1204. 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), 4138-4151. doi: 10.1016/j.na.2011.03.047.  Google Scholar

[29]

Z. P. Xin, On nonlinear stability of contact discontinuities, in "Hyperbolic Problems: Theory, Numerics, Applications" (Stony Brook, NY, 1994), World Sci. Publ., River Edge, NJ, (1996), 249-257.  Google Scholar

show all references

References:
[1]

D. G. Aronson, The porous media equation, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Math., 1224, Springer, Berlin, (1986), 1-46.  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), 511-541. 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), 155-168. 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), 55-77. 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), 1246-1297. 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), 89-116. 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), 368-384. 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 "Analysis of Systems of Conservation Laws" (Aachen, 1997), Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 99, Chapman & Hall/CRC, Boca Raton, FL, (1999), 87-127.  Google Scholar

[9]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117. 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), 297-329. 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), 211-250. 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), 2601-2640. 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), 83-94. 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), 625-628.  Google Scholar

[15]

C. J. Lin, Asymptotic stability of rarefaction waves in radiative hydrodynamics, Comm. Math. Sci., 9 (2011), 207-223. 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), 451-465. 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), 34-84.  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), 1-13. 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), 428-453. 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), 1561-1597. 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), 2076-2110. doi: 10.1016/j.jde.2010.07.029.  Google Scholar

[23]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 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, New York, 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), 1030-1055. 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), 1189-1204. 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), 4138-4151. doi: 10.1016/j.na.2011.03.047.  Google Scholar

[29]

Z. P. Xin, On nonlinear stability of contact discontinuities, in "Hyperbolic Problems: Theory, Numerics, Applications" (Stony Brook, NY, 1994), World Sci. Publ., River Edge, NJ, (1996), 249-257.  Google Scholar

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