This paper is devoted to the analysis of the Cauchy problem for a system of PDEs arising in radiative hydrodynamics. This system, which comes from the so-called equilibrium diffusion regime, is a variant of the usual Euler equations, where the energy and pressure functionals are modified to take into account the effect of radiation and the energy balance containing a nonlinear diffusion term acting on the temperature. The problem is studied in the multi-dimensional framework. The authors identify the existence of a strictly convex entropy and a stability property of the system, and check that the Kawashima-Shizuta condition holds. Then, based on these structure properties, the well-posedness close to a constant state can be proved by using fine energy estimates. The asymptotic decay of the solutions are also investigated.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Buet, C. and Després, B., Asymptotic preserving and positive schemes for radiation hydrodynamics, J. Comput. Phys., 215, 2006, 717–740.
Coulombel, J.-F., Golse, F. and Goudon, T., Diffusion approximation and entropy-based moment closure for kinetic equations, Asymptot. Anal., 45(1–2), 2005, 1–39.
Coulombel, J.-F. and Goudon, T., The strong relaxation limit of the multidimensional isothermal Euler equations (electronic), Trans. Amer. Math. Soc., 359(2),2007, 637–648.
Friedrichs, K. O., Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7, 1954, 345–392.
Godillon-Lafitte, P. and Goudon, T., A coupled model for radiative transfer: Doppler effects, equilibrium, and nonequilibrium diffusion asymptotics (electronic), Multiscale Model. Simul., 4(4), 2005, 1245–1279.
Goudon, T., Morel, J.-E., Després, B., et al., Mathematical Models and Numerical Methods for Radiative Transfer, Panoramas et Synthèses, 28, SMF, Paris, 2009.
Hanouzet, B. and Natalini, R., Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal., 169(2), 2003, 89–117.
Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58(3), 1975, 181–205.
Kawashima, S., Nikkuni, Y. and Nishibata, S., The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics, Analysis of Systems of Conservation Laws (Aachen, 1997), Monogr. Surv. Pure Appl. Math., 99, Chapman & Hall/CRC, Boca Raton, 1999, 87–127.
Kawashima, S., Nikkuni, Y. and Nishibata, S., Large-time behavior of solutions to hyperbolic-elliptic coupled systems, Arch. Ration. Mech. Anal., 170(4), 2003, 297–329.
Kawashima, S. and Yong, W.-A., Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal., 174(3), 2004, 345–364.
Lin, C., Mod`eles Mathématiques de la Théorie du Transfert Radiatif, Ph. D. Thesis, Université des Sciences et Technologies de Lille, 2007. Available at http://tel.archives-ouvertes.fr/tel-00411849/fr/
Lin, C., Coulombel, J.-F. and Goudon, T., Asymptotic stability of shock profiles in radiative hydrodynamics, C. R. Math. Acad. Sci. Paris, 345(11), 2007, 625–628.
Lowrie, L., Morel, J. E. and Hittinger, J. A., The coupling of radiation and hydrodynamics, The Astrophysical J., 521, 1999, 432–450.
Majda, A., Compressible fluid flow and systems of conservation laws in several space variables, Appl. Math. Sci., 53, Springer-Verlag, New York, 1984.
Majda, A., Systems of conservation laws in several space variables, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, 1217–1224.
Matsumura, A. and Nishida, T., The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55(9), 1979, 337–342.
Matsumura, A. and Nishida, T., The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20(1), 1980, 67–104.
Mihalas, D. and Mihalas, B. W., Foundations of Radiation Hydrodynamics, Oxford University Press, New York, 1984.
Nishida, T., Nonlinear hyperbolic equations and related topics in fluid dynamics, Publications Mathématiques d’Orsay, No. 78-02, Département de Mathématique, Université de Paris-Sud, Orsay, 1978.
Shizuta, Y. and Kawashima, S., Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14(2), 1985, 249–275.
Yong, W.-A., Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal., 172(2), 2004, 247–266.
Project supported by the Fundamental Research Funds for the Central Universities (No. 2009B27514) and the National Natural Science Foundation of China (No. 10871059).
About this article
Cite this article
Lin, C., Goudon, T. Global existence of the equilibrium diffusion model in radiative hydrodynamics. Chin. Ann. Math. Ser. B 32, 549–568 (2011). https://doi.org/10.1007/s11401-011-0658-z
- Radiative hydrodynamics
- Initial value problem
- Equilibrium diffusion regime
- Energy method
2000 MR Subject Classification