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April  2017, 37(4): 2045-2063. doi: 10.3934/dcds.2017087

Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics

Peng Jiang 1,2,

1. 

Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Received  May 2016 Revised  November 2016 Published  December 2016

We are concerned with the global well-posedness of the diffusion approximation model in radiation hydrodynamics, which describe the compressible fluid dynamics taking into account the radiation effect under the non-local thermal equilibrium case. The model consist of the compressible Navier-Stokes equations coupled with the radiative transport equation with non-local terms. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The existence of global solution is proved based on the classical energy estimates, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a finite time although there is a complex interaction between photons and matter.

Keywords: Radiation hydrodynamics, diffusion approximation, compressible Navier-Stokes equations, global well-posedness, rate of convergence.
Mathematics Subject Classification: 76N10, 78A40.
Citation: Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087
References:
[1]

J. W. Bond, K. M. Watson and J. A. Welch, Atomic Theory of Gas Dynamics, Addison-Wesley, Rewading, Massachusetts, 1965.

[2]

C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 85 (2004), 385-418.  doi: 10.1016/S0022-4073(03)00233-4.

[3]

J. A. Carrillo, R. Duan and A. Moussa, Global classical solution close to equillibrium to the Vlasov-Euler-Fokker-Planck system, Kinet. Relat. Model, 4 (2011), 227-258.  doi: 10.3934/krm.2011.4.227.

[4]

J. I. Castor, Radiation Hydrodynamics Cambridge University Press, 2004. doi: 10.1017/CBO9780511536182.

[5]

B. Ducomet and E. Feireisl, The equation of magnetohydrodynamics: On the interation between matter and radiation in the evlution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.

[6]

B. Ducomet, E. Feireisl and S. Necasova, On a model in radiation hydrodynamics, Ann. Inst. H. Poincar Anal. Non Linaire, 28 (2011), 797-812.  doi: 10.1016/j.anihpc.2011.06.002.

[7]

Th. Goudon and P. Lafitte, A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, SIAM Multiscale Model. Simul., 4 (2005), 1245-1279.  doi: 10.1137/040621041.

[8]

X. Hu and D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.  doi: 10.1016/j.jde.2010.10.017.

[9]

X. Hu and D. Wang, Strong solutions to the three-dimensional compressible viscoelastic fluids, J. Differential Equation, 252 (2012), 4027-4067.  doi: 10.1016/j.jde.2011.11.021.

[10]

D. Hoff and K. Zumbrum, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. J, 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003.

[11]

E. Hopf, Mathematical Problems of Radiative Equilibrium, Stechert-Hafner, New York, 1964.

[12]

S. Jiang, F. C. Li and F. Xie, Nonrelativistic limits of the compressible Navier-Stokes-FourierP1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal., 47 (2015), 3726-3746.  doi: 10.1137/140987596.

[13]

S. Jiang, F. Xie and J. W. Zhang, A global existence result in radiation hydrodynamics, Industrial and Applied Mathematics in China, Series in Contemporary Applied Mathematics, High Edu. Press and World Scientific. Beijing, Singapore, 10 (2009), 25–48.

[14]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984.

[15]

S. Kawashima and S. Nishibata, A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics, Indiana Univ. Math. J., 50 (2001), 567-589.  doi: 10.1512/iumj.2001.50.1797.

[16]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gases, SIAM J. Math. Anal., 30 (1999), 95-117.  doi: 10.1137/S0036141097322169.

[17]

R. Kippenhahn and A. Weigert, Stellar Structure and Evolution Springer Verlag, Berlin-Heidelberg, 1994.

[18]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math, 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[19]

C. Lin, Asymptotic stability of rarefaction waves in radiation hydrodynamics, Comm. Math. Sci., 9 (2011), 207-223.  doi: 10.4310/CMS.2011.v9.n1.a10.

[20]

C. Lin, J. F. Coulombel and Th. Goudon, Shock profiles for non equilibrium radiating gases, Physica D, 218 (2006), 83-94.  doi: 10.1016/j.physd.2006.04.012.

[21]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Jpn. Acad. Ser. A, 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337.

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat conductive gases, J. Math. Kyoto Univ, 20 (1980), 67-104. 

[23]

D. Mihalas and B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics, Oxford University Press, 1984.

[24]

S. S. Penner and D. B. Olfe, Radiation and Reentry, Academic Press, New York, 1968.

[25]

G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, 1973.

[26]

C. Rohde and W.-A. Yong, The nonrelativistic limit in radiation hydrodynamics: I. Weak entropy solutions for a model problem, J. Diff. Eqns., 234 (2007), 91-109.  doi: 10.1016/j.jde.2006.11.010.

[27]

R. N. Thomas, Some Aspects of Non-Equilibrium Thermodynamics in the Presence of a Radiation Field, University of Colorado Press, Boulder, Colorado, 1965.

[28]

W. J. Wang and F. Xie, The initial value problem for a multi-dimensional radiation hydrodynamics model with viscosity,, Math. Methods Appl. Sci., 34 (2011), 776-791.  doi: 10.1002/mma.1398.

[29]

Y. B. Zeldovich and Y. P. Raizer, Phsics of Shock Waves and High-Temperture Hydrodynamic Phenomenon, Academic Press, 1966.

[30]

X. Zhong and S. Jiang, Local existence and finite time blow-up in multidimensional radiation hydrodynamics,, J. Math. Fluid Mech., 9 (2007), 543-564.  doi: 10.1007/s00021-005-0213-3.

show all references

References:
[1]

J. W. Bond, K. M. Watson and J. A. Welch, Atomic Theory of Gas Dynamics, Addison-Wesley, Rewading, Massachusetts, 1965.

[2]

C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 85 (2004), 385-418.  doi: 10.1016/S0022-4073(03)00233-4.

[3]

J. A. Carrillo, R. Duan and A. Moussa, Global classical solution close to equillibrium to the Vlasov-Euler-Fokker-Planck system, Kinet. Relat. Model, 4 (2011), 227-258.  doi: 10.3934/krm.2011.4.227.

[4]

J. I. Castor, Radiation Hydrodynamics Cambridge University Press, 2004. doi: 10.1017/CBO9780511536182.

[5]

B. Ducomet and E. Feireisl, The equation of magnetohydrodynamics: On the interation between matter and radiation in the evlution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.

[6]

B. Ducomet, E. Feireisl and S. Necasova, On a model in radiation hydrodynamics, Ann. Inst. H. Poincar Anal. Non Linaire, 28 (2011), 797-812.  doi: 10.1016/j.anihpc.2011.06.002.

[7]

Th. Goudon and P. Lafitte, A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, SIAM Multiscale Model. Simul., 4 (2005), 1245-1279.  doi: 10.1137/040621041.

[8]

X. Hu and D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231.  doi: 10.1016/j.jde.2010.10.017.

[9]

X. Hu and D. Wang, Strong solutions to the three-dimensional compressible viscoelastic fluids, J. Differential Equation, 252 (2012), 4027-4067.  doi: 10.1016/j.jde.2011.11.021.

[10]

D. Hoff and K. Zumbrum, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. J, 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003.

[11]

E. Hopf, Mathematical Problems of Radiative Equilibrium, Stechert-Hafner, New York, 1964.

[12]

S. Jiang, F. C. Li and F. Xie, Nonrelativistic limits of the compressible Navier-Stokes-FourierP1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal., 47 (2015), 3726-3746.  doi: 10.1137/140987596.

[13]

S. Jiang, F. Xie and J. W. Zhang, A global existence result in radiation hydrodynamics, Industrial and Applied Mathematics in China, Series in Contemporary Applied Mathematics, High Edu. Press and World Scientific. Beijing, Singapore, 10 (2009), 25–48.

[14]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984.

[15]

S. Kawashima and S. Nishibata, A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics, Indiana Univ. Math. J., 50 (2001), 567-589.  doi: 10.1512/iumj.2001.50.1797.

[16]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gases, SIAM J. Math. Anal., 30 (1999), 95-117.  doi: 10.1137/S0036141097322169.

[17]

R. Kippenhahn and A. Weigert, Stellar Structure and Evolution Springer Verlag, Berlin-Heidelberg, 1994.

[18]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math, 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[19]

C. Lin, Asymptotic stability of rarefaction waves in radiation hydrodynamics, Comm. Math. Sci., 9 (2011), 207-223.  doi: 10.4310/CMS.2011.v9.n1.a10.

[20]

C. Lin, J. F. Coulombel and Th. Goudon, Shock profiles for non equilibrium radiating gases, Physica D, 218 (2006), 83-94.  doi: 10.1016/j.physd.2006.04.012.

[21]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Jpn. Acad. Ser. A, 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337.

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat conductive gases, J. Math. Kyoto Univ, 20 (1980), 67-104. 

[23]

D. Mihalas and B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics, Oxford University Press, 1984.

[24]

S. S. Penner and D. B. Olfe, Radiation and Reentry, Academic Press, New York, 1968.

[25]

G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, 1973.

[26]

C. Rohde and W.-A. Yong, The nonrelativistic limit in radiation hydrodynamics: I. Weak entropy solutions for a model problem, J. Diff. Eqns., 234 (2007), 91-109.  doi: 10.1016/j.jde.2006.11.010.

[27]

R. N. Thomas, Some Aspects of Non-Equilibrium Thermodynamics in the Presence of a Radiation Field, University of Colorado Press, Boulder, Colorado, 1965.

[28]

W. J. Wang and F. Xie, The initial value problem for a multi-dimensional radiation hydrodynamics model with viscosity,, Math. Methods Appl. Sci., 34 (2011), 776-791.  doi: 10.1002/mma.1398.

[29]

Y. B. Zeldovich and Y. P. Raizer, Phsics of Shock Waves and High-Temperture Hydrodynamic Phenomenon, Academic Press, 1966.

[30]

X. Zhong and S. Jiang, Local existence and finite time blow-up in multidimensional radiation hydrodynamics,, J. Math. Fluid Mech., 9 (2007), 543-564.  doi: 10.1007/s00021-005-0213-3.

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