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December  2015, 8(4): 725-763. doi: 10.3934/krm.2015.8.725

Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime

1. 

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France

Received  March 2015 Revised  June 2015 Published  July 2015

We consider the relativistic transfer equations for photons interacting via emission absorption and scattering with a moving fluid. We prove a comparison principle and we study the non-equilibrium regime: the relativistic correction terms in the scattering operator lead to a frequency drift term modeling the Doppler effects. We prove that the solution of the relativistic transfer equations converges toward the solution of this drift diffusion equation.
Citation: Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic and Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725
References:
[1]

G. Allaire and F. Golse, Transport et Diffusion, Cours de l'école Polytechnique, chapitre 4, 2012.

[2]

C. Bardos, F. Golse and B. Perthame, The Rosseland approximation for the radiative transfer equations, Communications on Pure and Applied Mathematics, 40 (1987), 691-721. doi: 10.1002/cpa.3160400603.

[3]

C. Bardos, F. Golse, B. Perthame and R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation, Journal of Functional Analysis, 77 (1988), 434-460. doi: 10.1016/0022-1236(88)90096-1.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.

[5]

C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, Journal of Quantitative Spectroscopy and Radiative Transfer, 85 (2004), 385-418. doi: 10.1016/S0022-4073(03)00233-4.

[6]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, American Journal of Mathematics, 93 (1971), 265-298. doi: 10.2307/2373376.

[7]

J. A. Carrillo, J. Rosado and F. Salvarani, 1D nonlinear Fokker Planck equations for fermions and bosons, Applied Mathematics Letters, 21 (2008), 148-154. doi: 10.1016/j.aml.2006.06.023.

[8]

R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique Masson, 1985.

[9]

C. Dogbe, The radiative transfer equations: Diffusion approximation under accretiveness and compactness assumptions, Computers & Mathematics with Applications, 42 (2001), 783-791. doi: 10.1016/S0898-1221(01)00198-5.

[10]

B. Ducomet and S. Nečasová, Diffusion limits in a model of radiative flow, Annali Dell' Universita di Ferrara, 61 (2015), 17-59. doi: 10.1007/s11565-014-0214-3.

[11]

P. Godillon-Lafitte and T. Goudon, A coupled model for radiative transfer: Doppler effects, equilibrium, and non-equilibrium diffusion asymptotics, Multiscale Model. Simul., 4 (2005), 1245-1279. doi: 10.1137/040621041.

[12]

F. Golse and B. Perthame, Generalized solutions of the radiative transfer equations in a singular case, Communications in Mathematical Physics, 106 (1986), 211-239. doi: 10.1007/BF01454973.

[13]

F. Golse and F. Salvarani, The Rosseland limit for radiative transfer in gray matter, hal-00268799, 2008.

[14]

D. Mihalas and B. Weibel Mihalas, Foundations of Radiation Hydrodynamics, Oxford University Press, New York, 1984. doi: 10.1063/1.2815048.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

G. C. Pomraning, The Equations of Radiation Hydrodynamics, Dover Publications 1973.

[17]

A. M. Winslow, Multifrequency-grey method for radiation diffusion with Compton scattering, Journal of Computational Physics, 117 (1995), 262-273.

[18]

Y. Li and S. Zhu, Existence results and blow-up criterion of compressible radiation hydrodynamic equations, Journal of Dynamics and Differential Equations, 2015, arXiv:1407.7830. doi: 10.1007/s10884-015-9455-9.

show all references

References:
[1]

G. Allaire and F. Golse, Transport et Diffusion, Cours de l'école Polytechnique, chapitre 4, 2012.

[2]

C. Bardos, F. Golse and B. Perthame, The Rosseland approximation for the radiative transfer equations, Communications on Pure and Applied Mathematics, 40 (1987), 691-721. doi: 10.1002/cpa.3160400603.

[3]

C. Bardos, F. Golse, B. Perthame and R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation, Journal of Functional Analysis, 77 (1988), 434-460. doi: 10.1016/0022-1236(88)90096-1.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.

[5]

C. Buet and B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, Journal of Quantitative Spectroscopy and Radiative Transfer, 85 (2004), 385-418. doi: 10.1016/S0022-4073(03)00233-4.

[6]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, American Journal of Mathematics, 93 (1971), 265-298. doi: 10.2307/2373376.

[7]

J. A. Carrillo, J. Rosado and F. Salvarani, 1D nonlinear Fokker Planck equations for fermions and bosons, Applied Mathematics Letters, 21 (2008), 148-154. doi: 10.1016/j.aml.2006.06.023.

[8]

R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique Masson, 1985.

[9]

C. Dogbe, The radiative transfer equations: Diffusion approximation under accretiveness and compactness assumptions, Computers & Mathematics with Applications, 42 (2001), 783-791. doi: 10.1016/S0898-1221(01)00198-5.

[10]

B. Ducomet and S. Nečasová, Diffusion limits in a model of radiative flow, Annali Dell' Universita di Ferrara, 61 (2015), 17-59. doi: 10.1007/s11565-014-0214-3.

[11]

P. Godillon-Lafitte and T. Goudon, A coupled model for radiative transfer: Doppler effects, equilibrium, and non-equilibrium diffusion asymptotics, Multiscale Model. Simul., 4 (2005), 1245-1279. doi: 10.1137/040621041.

[12]

F. Golse and B. Perthame, Generalized solutions of the radiative transfer equations in a singular case, Communications in Mathematical Physics, 106 (1986), 211-239. doi: 10.1007/BF01454973.

[13]

F. Golse and F. Salvarani, The Rosseland limit for radiative transfer in gray matter, hal-00268799, 2008.

[14]

D. Mihalas and B. Weibel Mihalas, Foundations of Radiation Hydrodynamics, Oxford University Press, New York, 1984. doi: 10.1063/1.2815048.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

G. C. Pomraning, The Equations of Radiation Hydrodynamics, Dover Publications 1973.

[17]

A. M. Winslow, Multifrequency-grey method for radiation diffusion with Compton scattering, Journal of Computational Physics, 117 (1995), 262-273.

[18]

Y. Li and S. Zhu, Existence results and blow-up criterion of compressible radiation hydrodynamic equations, Journal of Dynamics and Differential Equations, 2015, arXiv:1407.7830. doi: 10.1007/s10884-015-9455-9.

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