Abstract
We consider a “semi-relativistic” model of radiative viscous compressible Navier–Stokes–Fourier system coupled to the radiative transfer equation extending the classical model introduced in Ducomet et al. (Ann Inst Henri Poincarè AN 28:797–812, 2011) and we study some of its singular limits (low Mach and diffusion) in the case of well-prepared initial data and Dirichlet boundary condition for the velocity field. In the low Mach number case we prove the convergence toward the incompressible Navier–Stokes system coupled to a system of two stationary transport equations. In the diffusion case we prove the convergence toward the compressible Navier–Stokes with modified state functions (equilibrium case) or toward the compressible Navier–Stokes coupled to a diffusion equation (non equilibrium case).
Similar content being viewed by others
References
Balian, R.: From Microphysics to Macrophysics. Methods and Applications of Statistical Physics, Vol. II. Springer, Berlin (1992)
Bardos C., Golse F., Perthame B.: The Rosseland approximation for the radiative transfer equation. Commun. Pure. Appl. Math. 40(6), 691–721 (1987)
Bardos C., Golse F., Perthame B., Sentis R.: The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation. J. Funct. Anal. 77(2), 434–460 (1988)
Berthon, C., Buet, C., Coulombel, J.-F., Desprès, B., Dubois, J., Goudon, T., Morel, J.E., Turpault, R.: Mathematical models and numerical methods for radiative transfer. In: Panoramas et Synthèses No. 28. Société Mathématique de France, Paris (2009)
Bournaveas N., Perthame B.: Averages over spheres for kinetic transport equations; hyperbolic Sobolev spaces and Strichartz inequalities. J. Math. Pures Appl. 80, 517–534 (2001)
Buet C., Després B.: Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics. J. Quant. Spectrosc. Radiat. Transf. 85, 385–480 (2004)
Dubroca B., Feugeas J.-L.: Etude théorique et numérique d’une hiérarchie de modéles aux moments pour le transfert radiatif. C. R. Acad. Sci. Paris 329, 915–920 (1999)
Dubroca B., Seaïd M., Feugeas J.-L.: A consistent approach for the coupling of radiation and hydrodynamics at low Mach number. J. Comput. Phys. 225, 1039–1065 (2007)
Ducomet B., Feireisl E., Nečasová Š.: On a model of radiation hydrodynamics. Ann. Inst. Henri Poincaré AN 28, 797–812 (2011)
Ducomet, B., Nečasová, Š.: Global smooth solution of the Cauchy problem for a model of radiative flow. Ann. Scuola Norm. Sup. Pisa (to appear)
Ducomet B., Nečasová Š.: Low Mach number limit for a model of radiative flow. J. Evol. Equ. 14, 357–385 (2014)
Ducomet, B., Nečasová, Š.: Diffusion limits in a model of radiative flow. Ann. Univ. Ferrara. doi:10.1007/s11565-014-0214-3 (to appear)
Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhauser, Basel 2009
Feireisl E., Novotný A.: Weak-strong uniqueness for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204(1), 683–706 (2012)
Golse F., Perthame B.: Generalized solutions of the radiative transfer equations in a singular case. Commun. Math. Phys. 106, 211–239 (1986)
Jiang S.: Global solutions of the Cauchy problem for a viscous polytropic ideal gas. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) XXVI, 47–74 (1998)
Levermore D.: Relating Eddington factors to flux limiters. J. Quant. Spectrosc. Radiat. Transf. 31, 149–160 (1984)
Levermore D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1076 (1996)
Ladyženskaja O.A., Solonnikov V.A., Ural’ceva N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS, New York (1968)
Lowrie R.B., Morel J.E., Hittinger J.A.: The coupling of radiation and hydrodynamics. Astrophys. J. 521, 432–450 (1999)
Matsumura, A., Nishida, T.: Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)
Mihalas B., Weibel-Mihalas B.: Foundations of Radiation Hydrodynamics. Dover Publications, Dover (1984)
Pomraning G.C.: Radiation Hydrodynamics. Dover Publications, New York (2005)
Ripoll J.F., Dubroca B., Duffa G.: Modelling radiative mean absorption coefficients. Combust. Theory Model. 5, 261–274 (2001)
Saint Raymond L.: Hydrodynamic limits: some improvements of the relative entropy method. Ann. Inst. Henri Poincaré AN 26, 705–744 (2009)
Teleaga I., Seaïd M., Gasser I., Klar A., Struckmeier J.: Radiation models for thermal flows at low Mach number. J. Comput. Phys. 215, 506–525 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Beirão da Veiga
Rights and permissions
About this article
Cite this article
Ducomet, B., Nečasová, Š. Singular Limits in a Model of Radiative Flow. J. Math. Fluid Mech. 17, 341–380 (2015). https://doi.org/10.1007/s00021-015-0204-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-015-0204-y