American Institute of Mathematical Sciences

Advanced
July  2015, 35(7): 3015-3037. doi: 10.3934/dcds.2015.35.3015

Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics

Peng Jiang 1,

1. 

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

Received  October 2013 Revised  March 2014 Published  January 2015

The governing equations in radiation hydrodynamics are derived from the conservation laws for macroscopic quantities, which have to be coupled with a radiative transfer equation to account for the radiative effects. In the present paper, we work with a mathematical model for the diffusion approximation of radiation hydrodynamics in the simplified framework of 1-D flows. We prove the existence, uniqueness and regularity of global solutions to an initial-boundary value problem with large data. The existence of global solution is proved by combining the local existence theorem with the global a priori 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: initial-boundary value problem, diffusion approximation, a priori estimates., compressible Navier-Stokes equations, Radiation hydrodynamics, global solution.
Mathematics Subject Classification: 76N10, 78A4.
Citation: Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015
References:
[1]

J. W. Bond, K. M. Watson and J. A. Welch, Atomic Theory of Gas Dynamics,, Addison-Wesley, (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.  doi: 10.1016/S0022-4073(03)00233-4.  Google Scholar

[3]

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

[4]

G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large data,, J. Diff. Eqns., 182 (2002), 344.  doi: 10.1006/jdeq.2001.4111.  Google Scholar

[5]

A. Dedner and C. Rohde, Numerical approximation of entropy solutions for hyperbolic integro-differential equations,, Numer. Math., 97 (2004), 441.  doi: 10.1007/s00211-003-0502-9.  Google Scholar

[6]

A. Dedner and C. Rohde, FV-schemes for a scalar model problem of radiation magneto-hydrodynamics,, in Finite Volumes for Complex Applications III (eds. R. Herbin and D. Kröner), (2002), 165.   Google Scholar

[7]

B. Ducomet and E. Feireisl, On the dynamics of gaseous stars,, Arch. Rational Mech. Anal., 174 (2004), 221.  doi: 10.1007/s00205-004-0326-5.  Google Scholar

[8]

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.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

[9]

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

[10]

B. Ducomet and A. Zlotnik, Lyapunov functional method for 1-D radiative and reactive viscous gas dynamics,, Arch. Rational Mech. Anal., 177 (2005), 185.  doi: 10.1007/s00205-005-0363-8.  Google Scholar

[11]

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

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Equations of Second Order,, Springer, (1994).   Google Scholar

[13]

K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas,, Quart. J. Mech. Appl. Math., 24 (1971), 155.   Google Scholar

[14]

M. A. Heaslet and B. S. Baldwin, Predictions of the structure of radiation-resisted shock waves,, Phys. Fluids, 6 (1963), 781.  doi: 10.1063/1.1706814.  Google Scholar

[15]

E. Hopf, Mathematical Problems of Radiative Equilibrium,, Stechert-Hafner, (1964).   Google Scholar

[16]

S. Jiang, F. Xie and J. W. Zhang, A global existence result in radiation hydrodynamics,, in Industrial and Applied Mathematics in China, (2009), 25.   Google Scholar

[17]

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

[18]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gases,, SIAM J. Math. Anal., 30 (1999), 95.   Google Scholar

[19]

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

[20]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial bounday value problems for one-dimensional equations of a viscous gas,, J. Appl. Math. Mech., 41 (1977), 273.   Google Scholar

[21]

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

[22]

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

[23]

G. M. Lieberman, Second Order Parabolic Differential Equations,, Word Scientific Publishing Co. Pte. Ltd., (1996).  doi: 10.1142/3302.  Google Scholar

[24]

C. Lin, J. F. Coulombel and Th. Goudon, Shock profiles for non equilibrium radiating gases,, Physica D, 218 (2006), 83.  doi: 10.1016/j.physd.2006.04.012.  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.  doi: 10.1137/S0036141001386908.  Google Scholar

[26]

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

[27]

T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary,, J. Diff Eqns., 65 (1986), 49.  doi: 10.1016/0022-0396(86)90041-0.  Google Scholar

[28]

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: 10.1142/S0218202500000598.  Google Scholar

[29]

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

[30]

S. S. Penner and D. B. Olfe, Radiation and Reentry,, Academic Press, (1968).   Google Scholar

[31]

C. Rohde and F. Xie, Global Existence and blowup phenomenon for a 1D radiation hydrodynamics model problem,, Math. Methods Appl. Sci., 35 (2012), 564.  doi: 10.1002/mma.1593.  Google Scholar

[32]

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.  doi: 10.1016/j.jde.2006.11.010.  Google Scholar

[33]

D. H. Sampson, Radiative constributions to energy and momentum transport in a gas,, Interscience, (1965).   Google Scholar

[34]

P. Secchi, On the motion of gaseous stars in the presence of radiation,, Commu. PDE, 15 (1990), 185.  doi: 10.1080/03605309908820683.  Google Scholar

[35]

V. A. Solonnikov and A. V. Kazhikhov, Existence theorems for the equations of motion of a compressible viscous fluid,, Ann. Rev. Fluid Mech., 13 (1981), 79.  doi: 10.1146/annurev.fl.13.010181.000455.  Google Scholar

[36]

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

[37]

M. Umehara and A. Tani, Global solution to the one-dimensional equations for a self- gravtating viscous radiative and reactive gas,, J. Diff. Eqns., 234 (2007), 439.  doi: 10.1016/j.jde.2006.09.023.  Google Scholar

[38]

Y. B. Zeldovich and Y. P. Raizer, Phsics of shock waves and high-temperture hydrodynamic phenomenon,, Academic Press, (1966).   Google Scholar

[39]

X. Zhong and S. Jiang, Local existence and finite time blow-up in multidimensional radiation hydrodynamics,, J. Math. Fluid Mech., 9 (2007), 543.  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, (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.  doi: 10.1016/S0022-4073(03)00233-4.  Google Scholar

[3]

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

[4]

G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large data,, J. Diff. Eqns., 182 (2002), 344.  doi: 10.1006/jdeq.2001.4111.  Google Scholar

[5]

A. Dedner and C. Rohde, Numerical approximation of entropy solutions for hyperbolic integro-differential equations,, Numer. Math., 97 (2004), 441.  doi: 10.1007/s00211-003-0502-9.  Google Scholar

[6]

A. Dedner and C. Rohde, FV-schemes for a scalar model problem of radiation magneto-hydrodynamics,, in Finite Volumes for Complex Applications III (eds. R. Herbin and D. Kröner), (2002), 165.   Google Scholar

[7]

B. Ducomet and E. Feireisl, On the dynamics of gaseous stars,, Arch. Rational Mech. Anal., 174 (2004), 221.  doi: 10.1007/s00205-004-0326-5.  Google Scholar

[8]

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.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

[9]

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

[10]

B. Ducomet and A. Zlotnik, Lyapunov functional method for 1-D radiative and reactive viscous gas dynamics,, Arch. Rational Mech. Anal., 177 (2005), 185.  doi: 10.1007/s00205-005-0363-8.  Google Scholar

[11]

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

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Equations of Second Order,, Springer, (1994).   Google Scholar

[13]

K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas,, Quart. J. Mech. Appl. Math., 24 (1971), 155.   Google Scholar

[14]

M. A. Heaslet and B. S. Baldwin, Predictions of the structure of radiation-resisted shock waves,, Phys. Fluids, 6 (1963), 781.  doi: 10.1063/1.1706814.  Google Scholar

[15]

E. Hopf, Mathematical Problems of Radiative Equilibrium,, Stechert-Hafner, (1964).   Google Scholar

[16]

S. Jiang, F. Xie and J. W. Zhang, A global existence result in radiation hydrodynamics,, in Industrial and Applied Mathematics in China, (2009), 25.   Google Scholar

[17]

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

[18]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gases,, SIAM J. Math. Anal., 30 (1999), 95.   Google Scholar

[19]

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

[20]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial bounday value problems for one-dimensional equations of a viscous gas,, J. Appl. Math. Mech., 41 (1977), 273.   Google Scholar

[21]

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

[22]

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

[23]

G. M. Lieberman, Second Order Parabolic Differential Equations,, Word Scientific Publishing Co. Pte. Ltd., (1996).  doi: 10.1142/3302.  Google Scholar

[24]

C. Lin, J. F. Coulombel and Th. Goudon, Shock profiles for non equilibrium radiating gases,, Physica D, 218 (2006), 83.  doi: 10.1016/j.physd.2006.04.012.  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.  doi: 10.1137/S0036141001386908.  Google Scholar

[26]

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

[27]

T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary,, J. Diff Eqns., 65 (1986), 49.  doi: 10.1016/0022-0396(86)90041-0.  Google Scholar

[28]

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: 10.1142/S0218202500000598.  Google Scholar

[29]

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

[30]

S. S. Penner and D. B. Olfe, Radiation and Reentry,, Academic Press, (1968).   Google Scholar

[31]

C. Rohde and F. Xie, Global Existence and blowup phenomenon for a 1D radiation hydrodynamics model problem,, Math. Methods Appl. Sci., 35 (2012), 564.  doi: 10.1002/mma.1593.  Google Scholar

[32]

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.  doi: 10.1016/j.jde.2006.11.010.  Google Scholar

[33]

D. H. Sampson, Radiative constributions to energy and momentum transport in a gas,, Interscience, (1965).   Google Scholar

[34]

P. Secchi, On the motion of gaseous stars in the presence of radiation,, Commu. PDE, 15 (1990), 185.  doi: 10.1080/03605309908820683.  Google Scholar

[35]

V. A. Solonnikov and A. V. Kazhikhov, Existence theorems for the equations of motion of a compressible viscous fluid,, Ann. Rev. Fluid Mech., 13 (1981), 79.  doi: 10.1146/annurev.fl.13.010181.000455.  Google Scholar

[36]

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

[37]

M. Umehara and A. Tani, Global solution to the one-dimensional equations for a self- gravtating viscous radiative and reactive gas,, J. Diff. Eqns., 234 (2007), 439.  doi: 10.1016/j.jde.2006.09.023.  Google Scholar

[38]

Y. B. Zeldovich and Y. P. Raizer, Phsics of shock waves and high-temperture hydrodynamic phenomenon,, Academic Press, (1966).   Google Scholar

[39]

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

[1]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[2]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[3]

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
[4]

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
[5]

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
[6]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[7]

Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
[8]

Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115
[9]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
[10]

Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201
[11]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[12]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[13]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
[14]

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
[15]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[16]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609
[17]

Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277
[18]

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
[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]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences