January  2011, 10(1): 209-223. doi: 10.3934/cpaa.2011.10.209

Asymptotic behavior of solutions to a model system of a radiating gas

1. 

Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

Received  January 2010 Revised  April 2010 Published  November 2010

In this paper we focus on the initial value problem for a hyperbolic-elliptic coupled system of a radiating gas in multi-dimensional space. By using a time-weighted energy method, we obtain the global existence and optimal decay estimates of solutions. Moreover, we show that the solution is asymptotic to the linear diffusion wave which is given in terms of the heat kernel.
Citation: Yongqin Liu, Shuichi Kawashima. Asymptotic behavior of solutions to a model system of a radiating gas. Communications on Pure & Applied Analysis, 2011, 10 (1) : 209-223. doi: 10.3934/cpaa.2011.10.209
References:
[1]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,", Interscience Publishers, (1948).

[2]

M. Di Francesco, Initial value problem and relaxation limits of the Hamer model for radiating gases in several space variables,, NoDEA Nonl. Differential Equations Appl., 13 (2007), 531. doi: doi:10.1007/s00030-006-4023-y.

[3]

M. Di Francesco, "Diffusive Behavior and Asymptotic Self similarity for Fluid Models,", Ph. D thesis, (2004).

[4]

M. Di Francesco and C. Lattanzio, Optimal $L^1$ rate of decay to diffusion waves for the Hamer model of radiating gases,, Appl. Math. Lett., 19 (2006), 1046. doi: doi:10.1016/j.aml.2004.11.008.

[5]

R. Duan, K. Fellner and C. Zhu, Energy method for multi-dimensional balance laws with non-local dissipation,, to appear in J. Math. Pur. Appl. (http://homepage.univie.ac.at/klemens.fellner/preprints/DZFfinal.pdf)., ().

[6]

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 $n$-dimensions,, J. Differential Equations, 244 (2008), 2614. doi: doi:10.1016/j.jde.2008.02.023.

[7]

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: doi:10.1142/S0218202508002760.

[8]

K. Hamer, Nonlinear effects on the propogation of sound waves in a radiating gas,, Quart. J. Mech. Appl. Math., 24 (1971), 155. doi: doi:10.1093/qjmam/24.2.155.

[9]

T. Iguchi and S. Kawashima, On space-time decay properties of solutions to hyperbolic-elliptic coupled systems,, Hiroshima Math. J., 32 (2002), 229.

[10]

S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics,, in, (1998), 87.

[11]

S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems,, Arch. Rational Mech. Anal., 170 (2003), 297. doi: doi:10.1007/s00205-003-0273-6.

[12]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gas,, SIAM J. Math. Anal., 30 (1999), 95. doi: doi:10.1137/S0036141097322169.

[13]

S. Kawashima and S. Nishibata, Weak solutions with a shock to a model system of the radiating gas,, Sci. Bull. Josai. Univ., 5 (1998), 119.

[14]

S. Kawashima and S. Nishibata, Cauchy problem for a model system of the radiating gas: weak solution with a jump and classical solutions,, Math. Models Methods Appl. Sci., 9 (1999), 69. doi: doi:10.1142/S0218202599000063.

[15]

S. Kawashima and S. Nishibata, A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics,, Indiana Univ. Math. J., 50 (2001), 567. doi: doi:10.1512/iumj.2001.50.1797.

[16]

S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas,, Kyushu J. Math., 58 (2004), 211. doi: doi:10.2206/kyushujm.58.211.

[17]

C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas,, J. Differential Equations, 190 (2003), 439. doi: doi:10.1016/S0022-0396(02)00158-4.

[18]

C. Lattanzio, C. Mascia, T. Nguyen, R.G. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles,, SIAM J. Math. Anal., 41 (): 2165. doi: doi:10.1137/09076026X.

[19]

C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems,, Indiana Univ. Math. J., 56 (2007), 2601. doi: doi:10.1512/iumj.2007.56.3043.

[20]

C. Lattanzio, C. Mascia and D. Serre, in "Hyperbolic Problems: Theory, Numerics, Applications", (Lyon, (1721), 661.

[21]

P. Laurencot, Asymptotic self-similarity for a simplified model for radiating gases,, Asymptot. Anal., 42 (2005), 251.

[22]

C. Lin, J.-F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases,, Phys. D, 218 (2006), 83. doi: doi:10.1016/j.physd.2006.04.012.

[23]

C. Lin, J.-F. Coulombel and T. Goudon, Asymptotic stability of shock profiles in radiative hydrodynamics,, C. R. Math. Acad. Sci. Paris, 345 (2007), 625.

[24]

Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,, preprint, ().

[25]

H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws,, SIAM J. Math. Anal., 33 (2001), 930.

[26]

T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems,, Phys. D, 239 (2010), 428. doi: doi:10.1016/j.physd.2010.01.011.

[27]

S. Nishibata, Asymptotic behavior of solutions to a model system of a radiating gas with discontinuous initial data,, Math. Models Methods Appl. Sci., 10 (2000), 1209. doi: doi:10.1142/S0218202500000598.

[28]

M. Nishikawa and S. Nishibata, Convergence rates toward the travelling waves for a model system of the radiating gas,, Math. Meth. Appl. Sci., 30 (2007), 649. doi: doi:10.1002/mma.800.

[29]

L. Z. Ruan and C. J. Zhu, Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas,, preprint., ().

[30]

S. Schochet and E. Tadmor, The regularized Chapman-Enskog expansion for scalar conservation laws,, Arch. Rational Mech. Anal., 119 (1992), 95. doi: doi:10.1007/BF00375117.

[31]

D. Serre, $L^1$-stability of constants in a model for radiating gases,, Comm. Math. Sci., 1 (2003), 197.

[32]

D. Serre, $L^1$-stability of nonlinear waves in scalar conservation laws,, in, ().

[33]

W. G. Vincenti and C. H. Kruger, "Introduction to Physical Gas Dynamics,", Wiley, (1965).

[34]

W. Wang and W. Wang, The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions,, Nonlinear Analysis, 71 (2009), 1180. doi: doi:10.1016/j.na.2008.11.050.

show all references

References:
[1]

R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,", Interscience Publishers, (1948).

[2]

M. Di Francesco, Initial value problem and relaxation limits of the Hamer model for radiating gases in several space variables,, NoDEA Nonl. Differential Equations Appl., 13 (2007), 531. doi: doi:10.1007/s00030-006-4023-y.

[3]

M. Di Francesco, "Diffusive Behavior and Asymptotic Self similarity for Fluid Models,", Ph. D thesis, (2004).

[4]

M. Di Francesco and C. Lattanzio, Optimal $L^1$ rate of decay to diffusion waves for the Hamer model of radiating gases,, Appl. Math. Lett., 19 (2006), 1046. doi: doi:10.1016/j.aml.2004.11.008.

[5]

R. Duan, K. Fellner and C. Zhu, Energy method for multi-dimensional balance laws with non-local dissipation,, to appear in J. Math. Pur. Appl. (http://homepage.univie.ac.at/klemens.fellner/preprints/DZFfinal.pdf)., ().

[6]

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 $n$-dimensions,, J. Differential Equations, 244 (2008), 2614. doi: doi:10.1016/j.jde.2008.02.023.

[7]

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: doi:10.1142/S0218202508002760.

[8]

K. Hamer, Nonlinear effects on the propogation of sound waves in a radiating gas,, Quart. J. Mech. Appl. Math., 24 (1971), 155. doi: doi:10.1093/qjmam/24.2.155.

[9]

T. Iguchi and S. Kawashima, On space-time decay properties of solutions to hyperbolic-elliptic coupled systems,, Hiroshima Math. J., 32 (2002), 229.

[10]

S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics,, in, (1998), 87.

[11]

S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems,, Arch. Rational Mech. Anal., 170 (2003), 297. doi: doi:10.1007/s00205-003-0273-6.

[12]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gas,, SIAM J. Math. Anal., 30 (1999), 95. doi: doi:10.1137/S0036141097322169.

[13]

S. Kawashima and S. Nishibata, Weak solutions with a shock to a model system of the radiating gas,, Sci. Bull. Josai. Univ., 5 (1998), 119.

[14]

S. Kawashima and S. Nishibata, Cauchy problem for a model system of the radiating gas: weak solution with a jump and classical solutions,, Math. Models Methods Appl. Sci., 9 (1999), 69. doi: doi:10.1142/S0218202599000063.

[15]

S. Kawashima and S. Nishibata, A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics,, Indiana Univ. Math. J., 50 (2001), 567. doi: doi:10.1512/iumj.2001.50.1797.

[16]

S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas,, Kyushu J. Math., 58 (2004), 211. doi: doi:10.2206/kyushujm.58.211.

[17]

C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas,, J. Differential Equations, 190 (2003), 439. doi: doi:10.1016/S0022-0396(02)00158-4.

[18]

C. Lattanzio, C. Mascia, T. Nguyen, R.G. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles,, SIAM J. Math. Anal., 41 (): 2165. doi: doi:10.1137/09076026X.

[19]

C. Lattanzio, C. Mascia and D. Serre, Shock waves for radiative hyperbolic-elliptic systems,, Indiana Univ. Math. J., 56 (2007), 2601. doi: doi:10.1512/iumj.2007.56.3043.

[20]

C. Lattanzio, C. Mascia and D. Serre, in "Hyperbolic Problems: Theory, Numerics, Applications", (Lyon, (1721), 661.

[21]

P. Laurencot, Asymptotic self-similarity for a simplified model for radiating gases,, Asymptot. Anal., 42 (2005), 251.

[22]

C. Lin, J.-F. Coulombel and T. Goudon, Shock profiles for non-equilibrium radiating gases,, Phys. D, 218 (2006), 83. doi: doi:10.1016/j.physd.2006.04.012.

[23]

C. Lin, J.-F. Coulombel and T. Goudon, Asymptotic stability of shock profiles in radiative hydrodynamics,, C. R. Math. Acad. Sci. Paris, 345 (2007), 625.

[24]

Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,, preprint, ().

[25]

H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws,, SIAM J. Math. Anal., 33 (2001), 930.

[26]

T. Nguyen, R. G. Plaza and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems,, Phys. D, 239 (2010), 428. doi: doi:10.1016/j.physd.2010.01.011.

[27]

S. Nishibata, Asymptotic behavior of solutions to a model system of a radiating gas with discontinuous initial data,, Math. Models Methods Appl. Sci., 10 (2000), 1209. doi: doi:10.1142/S0218202500000598.

[28]

M. Nishikawa and S. Nishibata, Convergence rates toward the travelling waves for a model system of the radiating gas,, Math. Meth. Appl. Sci., 30 (2007), 649. doi: doi:10.1002/mma.800.

[29]

L. Z. Ruan and C. J. Zhu, Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas,, preprint., ().

[30]

S. Schochet and E. Tadmor, The regularized Chapman-Enskog expansion for scalar conservation laws,, Arch. Rational Mech. Anal., 119 (1992), 95. doi: doi:10.1007/BF00375117.

[31]

D. Serre, $L^1$-stability of constants in a model for radiating gases,, Comm. Math. Sci., 1 (2003), 197.

[32]

D. Serre, $L^1$-stability of nonlinear waves in scalar conservation laws,, in, ().

[33]

W. G. Vincenti and C. H. Kruger, "Introduction to Physical Gas Dynamics,", Wiley, (1965).

[34]

W. Wang and W. Wang, The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions,, Nonlinear Analysis, 71 (2009), 1180. doi: doi:10.1016/j.na.2008.11.050.

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