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

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.
Citation: Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete and Continuous Dynamical Systems, 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, 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. I. Castor, Radiation Hydrodynamics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511536182.

[4]

G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large data, J. Diff. Eqns., 182 (2002), 344-376. doi: 10.1006/jdeq.2001.4111.

[5]

A. Dedner and C. Rohde, Numerical approximation of entropy solutions for hyperbolic integro-differential equations, Numer. Math., 97 (2004), 441-471. doi: 10.1007/s00211-003-0502-9.

[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), Hermes Science Publications, 2002, 165-172.

[7]

B. Ducomet and E. Feireisl, On the dynamics of gaseous stars, Arch. Rational Mech. Anal., 174 (2004), 221-266. doi: 10.1007/s00205-004-0326-5.

[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), 595-629. doi: 10.1007/s00220-006-0052-y.

[9]

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.

[10]

B. Ducomet and A. Zlotnik, Lyapunov functional method for 1-D radiative and reactive viscous gas dynamics, Arch. Rational Mech. Anal., 177 (2005), 185-229. doi: 10.1007/s00205-005-0363-8.

[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), 1245-1279. doi: 10.1137/040621041.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Equations of Second Order, Springer, Berlin, 1994.

[13]

K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas, Quart. J. Mech. Appl. Math., 24 (1971), 155-168.

[14]

M. A. Heaslet and B. S. Baldwin, Predictions of the structure of radiation-resisted shock waves, Phys. Fluids, 6 (1963), 781-791. doi: 10.1063/1.1706814.

[15]

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

[16]

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

[17]

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.

[18]

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

[19]

S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math., 58 (2004), 211-250. doi: 10.2206/kyushujm.58.211.

[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), 273-282.

[21]

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

[22]

C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Diff. Eqns., 190 (2003), 439-465. doi: 10.1016/S0022-0396(02)00158-4.

[23]

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

[24]

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.

[25]

H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math. Anal., 33 (2001), 930-945. doi: 10.1137/S0036141001386908.

[26]

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

[27]

T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary, J. Diff Eqns., 65 (1986), 49-67. doi: 10.1016/0022-0396(86)90041-0.

[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), 1209-1231. doi: 10.1142/S0218202500000598.

[29]

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

[30]

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

[31]

C. Rohde and F. Xie, Global Existence and blowup phenomenon for a 1D radiation hydrodynamics model problem, Math. Methods Appl. Sci., 35 (2012), 564-573. doi: 10.1002/mma.1593.

[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), 91-109. doi: 10.1016/j.jde.2006.11.010.

[33]

D. H. Sampson, Radiative constributions to energy and momentum transport in a gas, Interscience, New York, 1965.

[34]

P. Secchi, On the motion of gaseous stars in the presence of radiation, Commu. PDE, 15 (1990), 185-204. doi: 10.1080/03605309908820683.

[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), 79-95. doi: 10.1146/annurev.fl.13.010181.000455.

[36]

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

[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), 439-463. doi: 10.1016/j.jde.2006.09.023.

[38]

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

[39]

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

[4]

G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohydrodynamics with large data, J. Diff. Eqns., 182 (2002), 344-376. doi: 10.1006/jdeq.2001.4111.

[5]

A. Dedner and C. Rohde, Numerical approximation of entropy solutions for hyperbolic integro-differential equations, Numer. Math., 97 (2004), 441-471. doi: 10.1007/s00211-003-0502-9.

[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), Hermes Science Publications, 2002, 165-172.

[7]

B. Ducomet and E. Feireisl, On the dynamics of gaseous stars, Arch. Rational Mech. Anal., 174 (2004), 221-266. doi: 10.1007/s00205-004-0326-5.

[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), 595-629. doi: 10.1007/s00220-006-0052-y.

[9]

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.

[10]

B. Ducomet and A. Zlotnik, Lyapunov functional method for 1-D radiative and reactive viscous gas dynamics, Arch. Rational Mech. Anal., 177 (2005), 185-229. doi: 10.1007/s00205-005-0363-8.

[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), 1245-1279. doi: 10.1137/040621041.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Equations of Second Order, Springer, Berlin, 1994.

[13]

K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas, Quart. J. Mech. Appl. Math., 24 (1971), 155-168.

[14]

M. A. Heaslet and B. S. Baldwin, Predictions of the structure of radiation-resisted shock waves, Phys. Fluids, 6 (1963), 781-791. doi: 10.1063/1.1706814.

[15]

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

[16]

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

[17]

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.

[18]

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

[19]

S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math., 58 (2004), 211-250. doi: 10.2206/kyushujm.58.211.

[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), 273-282.

[21]

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

[22]

C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Diff. Eqns., 190 (2003), 439-465. doi: 10.1016/S0022-0396(02)00158-4.

[23]

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

[24]

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.

[25]

H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math. Anal., 33 (2001), 930-945. doi: 10.1137/S0036141001386908.

[26]

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

[27]

T. Nagasawa, On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary, J. Diff Eqns., 65 (1986), 49-67. doi: 10.1016/0022-0396(86)90041-0.

[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), 1209-1231. doi: 10.1142/S0218202500000598.

[29]

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

[30]

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

[31]

C. Rohde and F. Xie, Global Existence and blowup phenomenon for a 1D radiation hydrodynamics model problem, Math. Methods Appl. Sci., 35 (2012), 564-573. doi: 10.1002/mma.1593.

[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), 91-109. doi: 10.1016/j.jde.2006.11.010.

[33]

D. H. Sampson, Radiative constributions to energy and momentum transport in a gas, Interscience, New York, 1965.

[34]

P. Secchi, On the motion of gaseous stars in the presence of radiation, Commu. PDE, 15 (1990), 185-204. doi: 10.1080/03605309908820683.

[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), 79-95. doi: 10.1146/annurev.fl.13.010181.000455.

[36]

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

[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), 439-463. doi: 10.1016/j.jde.2006.09.023.

[38]

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

[39]

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.

[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 and 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 and Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917

[4]

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

[5]

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

[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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115

[9]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5383-5405. doi: 10.3934/dcdsb.2020348

[10]

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

[11]

Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (126)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]