# American Institute of Mathematical Sciences

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). Google Scholar [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. Google Scholar [3] M. Di Francesco, "Diffusive Behavior and Asymptotic Self similarity for Fluid Models,", Ph. D thesis, (2004). Google Scholar [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. Google Scholar [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)., (). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] T. Iguchi and S. Kawashima, On space-time decay properties of solutions to hyperbolic-elliptic coupled systems,, Hiroshima Math. J., 32 (2002), 229. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] C. Lattanzio, C. Mascia and D. Serre, in "Hyperbolic Problems: Theory, Numerics, Applications", (Lyon, (1721), 661. Google Scholar [21] P. Laurencot, Asymptotic self-similarity for a simplified model for radiating gases,, Asymptot. Anal., 42 (2005), 251. Google Scholar [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. Google Scholar [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. Google Scholar [24] Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,, preprint, (). Google Scholar [25] H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws,, SIAM J. Math. Anal., 33 (2001), 930. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] L. Z. Ruan and C. J. Zhu, Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas,, preprint., (). Google Scholar [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. Google Scholar [31] D. Serre, $L^1$-stability of constants in a model for radiating gases,, Comm. Math. Sci., 1 (2003), 197. Google Scholar [32] D. Serre, $L^1$-stability of nonlinear waves in scalar conservation laws,, in, (). Google Scholar [33] W. G. Vincenti and C. H. Kruger, "Introduction to Physical Gas Dynamics,", Wiley, (1965). Google Scholar [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. Google Scholar

show all references

##### References:
 [1] R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,", Interscience Publishers, (1948). Google Scholar [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. Google Scholar [3] M. Di Francesco, "Diffusive Behavior and Asymptotic Self similarity for Fluid Models,", Ph. D thesis, (2004). Google Scholar [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. Google Scholar [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)., (). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] T. Iguchi and S. Kawashima, On space-time decay properties of solutions to hyperbolic-elliptic coupled systems,, Hiroshima Math. J., 32 (2002), 229. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] C. Lattanzio, C. Mascia and D. Serre, in "Hyperbolic Problems: Theory, Numerics, Applications", (Lyon, (1721), 661. Google Scholar [21] P. Laurencot, Asymptotic self-similarity for a simplified model for radiating gases,, Asymptot. Anal., 42 (2005), 251. Google Scholar [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. Google Scholar [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. Google Scholar [24] Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,, preprint, (). Google Scholar [25] H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws,, SIAM J. Math. Anal., 33 (2001), 930. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] L. Z. Ruan and C. J. Zhu, Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas,, preprint., (). Google Scholar [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. Google Scholar [31] D. Serre, $L^1$-stability of constants in a model for radiating gases,, Comm. Math. Sci., 1 (2003), 197. Google Scholar [32] D. Serre, $L^1$-stability of nonlinear waves in scalar conservation laws,, in, (). Google Scholar [33] W. G. Vincenti and C. H. Kruger, "Introduction to Physical Gas Dynamics,", Wiley, (1965). Google Scholar [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. Google Scholar
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