On a model in radiation hydrodynamics
Ducomet, Bernard ; Feireisl, Eduard ; Nečasová, Šárka
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011), p. 797-812 / Harvested from Numdam
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We consider a simplified model arising in radiation hydrodynamics based on the Navier–Stokes–Fourier system describing the macroscopic fluid motion, and a transport equation modeling the propagation of radiative intensity. We establish global-in-time existence for the associated initial–boundary value problem in the framework of weak solutions.

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     author = {Ducomet, Bernard and Feireisl, Eduard and Ne\v casov\'a, \v S\'arka},
     title = {On a model in radiation hydrodynamics},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {28},
     year = {2011},
     pages = {797-812},
     doi = {10.1016/j.anihpc.2011.06.002},
     mrnumber = {2859928},
     zbl = {1328.76074},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2011__28_6_797_0}
}

Ducomet, Bernard; Feireisl, Eduard; Nečasová, Šárka. On a model in radiation hydrodynamics. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) pp. 797-812. doi : 10.1016/j.anihpc.2011.06.002. http://gdmltest.u-ga.fr/item/AIHPC_2011__28_6_797_0/

[1] A.A. Amosov, Well-posedness “in the large” initial and boundary-value problems for the system of dynamical equations of a viscous radiating gas, Sov. Physics Dokl. 30 (1985), 129-131 | Zbl 0654.35019

[2] C. Bardos, F. Golse, B. Perthame, R. Sentis, The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation, J. Funct. Anal. 77 (1988), 434-460 | MR 933978 | Zbl 0655.35075

[3] M.E. Bogovskii, Solution of some vector analysis problems connected with operators div and grad, Trudy Sem. S.L. Sobolev 80 no. 1 (1980), 5-40 | MR 631691

[4] N. Bournaveas, B. Perthame, Averages over spheres for kinetic transport equations; hyperbolic Sobolev spaces and Strichartz inequalities, J. Math. Pures Appl. 80 no. 9 (2001), 517-534 | MR 1831433 | Zbl 1036.82023

[5] C. Buet, B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectroscopy Rad. Transf. 85 (2004), 385-480

[6] S. Chandrasekhar, Radiative Transfer, Dover Publications, New York (1960) | MR 111583 | Zbl 0037.43201

[7] R. Coifman, Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315-331 | MR 380244 | Zbl 0324.44005

[8] R. Dautray, J.P. Watteau (ed.), La fusion thermonucléaire inertielle par laser, Eyrolles, Paris (1993)

[9] R.J. Diperna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511-547 | MR 1022305 | Zbl 0696.34049

[10] B. Dubroca, J.-L. Feugeas, Etude théorique et numérique dʼune hiérarchie de modéles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris 329 (1999), 915-920 | MR 1728008

[11] B. Ducomet, E. Feireisl, On the dynamics of gaseous stars, Arch. Ration. Mech. Anal. 174 (2004), 221-266 | MR 2098107 | Zbl 1085.76061

[12] B. Ducomet, E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. 266 (2006), 595-629 | MR 2238891 | Zbl 1113.76098

[13] B. Ducomet, Š. Nečasová, Global existence of solutions for the one-dimensional motions of a compressible gas with radiation: an “infrarelativistic model”, Nonlinear Anal. 72 (2010), 3258-3274 | MR 2587361 | Zbl 1185.35158

[14] B. Ducomet, Š. Nečasová, Global weak solutions to the 1D compressible Navier–Stokes equations with radiation, Commun. Math. Anal. 8 (2010), 23-65 | MR 2738332 | Zbl 1194.35305

[15] S. Eliezer, A. Ghatak, H. Hora, An Introduction to Equations of States, Theory and Applications, Cambridge University Press, Cambridge (1986)

[16] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford (2001) | MR 2040667

[17] E. Feireisl, A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser, Basel (2009) | MR 2499296 | Zbl 1176.35126

[18] E. Feireisl, On the motion of a viscous, compressible, and heat conducting fluid, Indiana Univ. Math. J. 53 (2004), 1707-1740 | MR 2106342

[19] E. Feireisl, On compactness of solutions to the compressible isentropic Navier–Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin. 42 no. 1 (2001), 83-98 | MR 1825374 | Zbl 1115.35096

[20] E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier–Stokes equations of compressible isentropic fluids, J. Math. Fluid Mech. 3 (2001), 358-392 | MR 1867887 | Zbl 0997.35043

[21] E. Feireisl, H. Petzeltová, On integrability up to the boundary of the weak solutions of the Navier–Stokes equations of compressible flow, Comm. Partial Differential Equations 25 no. 3-4 (2000), 755-767 | MR 1748351 | Zbl 0966.35096

[22] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, I, Springer-Verlag, New York (1994) | MR 1284206

[23] F. Golse, B. Perthame, Generalized solutions of the radiative transfer equations in a singular case, Comm. Math. Phys. 106 no. 2 (1986), 211-239 | MR 855310 | Zbl 0614.35084

[24] F. Golse, P.L. Lions, B. Perthame, R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 16 (1988), 110-125 | MR 923047 | Zbl 0652.47031

[25] F. Golse, B. Perthame, R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale dʼun opérateur de transport, C. R. Acad. Sci. Paris 301 (1985), 341-344 | MR 808622 | Zbl 0591.45007

[26] P. Jiang, D. Wang, Formation of singularities of solutions of the radiative transfer equations in a singular case, preprint, March 11, 2009.

[27] P. Jiang, D. Wang, Global weak solutions to the Euler–Boltzmann equations in radiation hydrodynamics, preprint, June 27, 2009. | MR 2920613

[28] C. Lin, Mathematical analysis of radiative transfer models, PhD thesis, 2007.

[29] C. Lin, J.F. Coulombel, T. Goudon, Shock profiles for non-equilibrium radiative gases, Phys. D 218 (2006), 83-94 | MR 2234210 | Zbl 1096.35086

[30] R.B. Lowrie, J.E. Morel, J.A. Hittinger, The coupling of radiation and hydrodynamics, Astrophys. J. 521 (1999), 432-450

[31] P.-L. Lions, Mathematical Topics in Fluid Dynamics, vol. 2: Compressible Models, Oxford Science Publication, Oxford (1998)

[32] P.-L. Lions, Bornes sur la densité pour les équations de Navier–Stokes compressible isentropiques avec conditions aux limites de Dirichlet, C. R. Acad. Sci. Paris 328 (1999), 659-662 | MR 1680813

[33] B. Mihalas, Stellar Atmospheres, W.H. Freeman and Cie (1978)

[34] B. Mihalas, B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics, Dover Publications, Dover (1984) | MR 781346 | Zbl 0651.76005

[35] A. Munier, R. Weaver, Radiation transfer in the fluid frame: a covariant formulation, Part I: Radiation hydrodynamics, Computer Phys. Rep. 3 (1986), 125-164

[36] A. Munier, R. Weaver, Radiation transfer in the fluid frame: a covariant formulation, Part II: Radiation transfer equation, Computer Phys. Rep. 3 (1986), 165-208

[37] P. Pedregal, Parametrized Measures and Variational Principles, Birkhäuser, Basel (1997) | MR 1452107 | Zbl 0879.49017

[38] G.C. Pomraning, Radiation Hydrodynamics, Dover Publications, New York (2005) | Zbl 0779.76084

[39] J.F. Ripoll, B. Dubroca, G. Duffa, Modelling radiative mean absorption coefficients, Combust. Theory Model. 5 (2001), 261-274 | Zbl 1021.78002

[40] T. Ruggeri, M. Trovato, Hyperbolicity in extended thermodynamics of Fermi and Bose gases, Contin. Mech. Thermodyn. 16 (2004), 551-576 | MR 2099281 | Zbl 1114.82018

[41] X. Zhong, J. Jiang, Local existence and finite-time blow up in multidimensional radiation hydrodynamics, J. Math. Fluid Mech. 9 (2007), 543-564 | MR 2374158 | Zbl 1133.76043