American Institute of Mathematical Sciences

Advanced
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

Yongqin Liu 1, and  Shuichi Kawashima 1,

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.
Keywords: Radiating gas, initial value problem, asymptotic behavior..
Mathematics Subject Classification: 35B40, 35M2.
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

[1]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[2]

Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319
[3]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431
[4]

Davide Bellandi. On the initial value problem for a class of discrete velocity models. Mathematical Biosciences & Engineering, 2017, 14 (1) : 31-43. doi: 10.3934/mbe.2017003
[5]

Masashi Ohnawa. Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities. Kinetic & Related Models, 2012, 5 (4) : 857-872. doi: 10.3934/krm.2012.5.857
[6]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151
[7]

Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212
[8]

Kai Yan, Zhaoyang Yin. On the initial value problem for higher dimensional Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1327-1358. doi: 10.3934/dcds.2015.35.1327
[9]

Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709
[10]

Fei Guo, Bao-Feng Feng, Hongjun Gao, Yue Liu. On the initial-value problem to the Degasperis-Procesi equation with linear dispersion. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1269-1290. doi: 10.3934/dcds.2010.26.1269
[11]

Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115
[12]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[13]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817
[14]

Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917
[15]

Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161
[16]

Patrizia Pucci, Maria Cesarina Salvatori. On an initial value problem modeling evolution and selection in living systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 807-821. doi: 10.3934/dcdss.2014.7.807
[17]

Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005
[18]

Yinnian He, Yi Li. Asymptotic behavior of linearized viscoelastic flow problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 843-856. doi: 10.3934/dcdsb.2008.10.843
[19]

Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052
[20]

Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences