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

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.

Citation: Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 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.Google Scholar

[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. Google Scholar

[3]

J. A. CarrilloR. 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. Google Scholar

[4]

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

[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. Google Scholar

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B. DucometE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

S. JiangF. 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. Google Scholar

[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. Google Scholar

[14]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984.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), 567-589. doi: 10.1512/iumj.2001.50.1797. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[20]

C. LinJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

[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. Google Scholar

[29]

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

[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. Google Scholar

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.Google Scholar

[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. Google Scholar

[3]

J. A. CarrilloR. 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. Google Scholar

[4]

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

[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. Google Scholar

[6]

B. DucometE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

S. JiangF. 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. Google Scholar

[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. Google Scholar

[14]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984.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), 567-589. doi: 10.1512/iumj.2001.50.1797. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[20]

C. LinJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

[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. Google Scholar

[29]

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

[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. Google Scholar

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