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

Thomas  Leroy

doi:10.3934/krm.2015.8.725

Article Contents

2015, Volume 8, Issue 4: 725-763. Doi: 10.3934/krm.2015.8.725

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

Thomas Leroy^1,

1.
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris Cedex 05

Received: February 28, 2015

Revised: May 31, 2015

Published: November 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B51, 35Q85, 35B40; Secondary: 35K65.

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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.