American Institute of Mathematical Sciences

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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

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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

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