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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 |
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. |
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