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Global Existence of a Weak Solution for a Model in Radiation Magnetohydrodynamics

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Abstract

We consider a simplified model based on the Navier-Stokes-Fourier system coupled to a transport equation and the Maxwell system, proposed to describe radiative flows in stars. We establish global-in-time existence for the associated initial-boundary value problem in the framework of weak solutions.

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Notes

  1. Naturally the \(T_{k}\) function has to be approximated by differentiable functions first.

References

  1. Allaire, G., Golse, F.: Transport et diffusion. Lecture Notes, Ecole Polytechnique

  2. Amosov, A.A.: Well-posedness “in the large” initial and boundary-value problems for the system of dynamical equations of a viscous radiating gas. Sov. Phys. Dokl. 30, 129–131 (1985)

    MATH  Google Scholar 

  3. Blanc, X., Ducomet, B.: Weak and strong solutions of equations of compressible magnetohydrodynamics. Preprint (2015)

  4. Bournaveas, N., Perthame, B.: Averages over spheres for kinetic transport equations; hyperbolic Sobolev spaces and Strichartz inequalities. J. Math. Pures Appl. 80(9), 517–534 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  Google Scholar 

  6. Coifman, R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cabannes, H.: Theoretical Magnetofluiddynamics. Academic Press, New York (1970)

    Google Scholar 

  8. DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MATH  Google Scholar 

  10. Ducomet, B., Feireisl, E.: The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars. Commun. Math. Phys. 266, 595–629 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ducomet, B., Feireisl, E., Nečasová, Š.: On a model in radiation hydrodynamics. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28, 797–812 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ducomet, B., Nečasová, Š.: Global existence of solutions for the one-dimensional motions of a compressible gas with radiation: An “infrarelativistic model”. Nonlinear Anal. 72, 3258–3274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ducomet, B., Nečasová, Š.: Global weak solutions to the 1D compressible Navier-Stokes equations with radiation. Commun. Math. Anal. 8, 23–65 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Ducomet, B., Nečasová, Š.: Large-time behavior of the motion of a viscous heat-conducting one-dimensional gas coupled to radiation. Ann. Mat. Pura Appl. 191, 219–260 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ducomet, B., Nečasová, Š.: Asymptotic behavior of the motion of a viscous heat-conducting one-dimensional gas with radiation: The pure scattering case. Anal. Appl. 11, 1350003 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ducomet, B., Nečasová, Š.: Global existence for weak solutions of the Cauchy problem in a model of radiation hydrodynamics. J. Math. Anal. Appl. 420, 464–482 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Heidelberg (1976)

    Book  MATH  Google Scholar 

  18. Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

  19. Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser, Basel (2009)

    Book  MATH  Google Scholar 

  20. Feireisl, E.: On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J. 53, 1707–1740 (2004)

    Article  MathSciNet  Google Scholar 

  21. Feireisl, E.: On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable. Comment. Math. Univ. Carol. 42, 83–98 (2001)

    MathSciNet  MATH  Google Scholar 

  22. Feireisl, E., Petzeltová, H.: On integrability up to the boundary of the weak solutions of the Navier-Stokes equations of compressible flow. Commun. Partial Differ. Equ. 25, 755–767 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Golse, F., Perthame, B.: Generalized solutions of the radiative transfer equations in a singular case. Commun. Math. Phys. 106, 211–239 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Golse, F., Lions, P.L., Perthame, B., Sentis, R.: Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 16, 110–125 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Golse, F., Perthame, B., Sentis, R.: Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport. C. R. Acad. Sci. Paris 301, 341–344 (1985)

    MathSciNet  MATH  Google Scholar 

  26. Jiang, P., Wang, D.: Formation of singularities of solutions of the radiative transfer equations in a singular case. Preprint (2009)

  27. Jiang, P., Wang, D.: Global weak solutions to the Euler-Boltzmann equations in radiation hydrodynamics. Preprint (June 27, 2009)

  28. Kolb, U.: Extreme Environment Astrophysics. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  29. Kulikovskiy, A.G., Lyubimov, G.A.: Magnetohydrodynamics. Addison-Wesley, Reading (1965)

    Google Scholar 

  30. Lin, C.: Mathematical analysis of radiative transfer models. PhD Thesis (2007)

  31. Lin, C., Coulombel, J.F., Goudon, T.: Shock profiles for non-equilibrium radiative gases. Physica D 218, 83–94 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lowrie, R.B., Morel, J.E., Hittinger, J.A.: The coupling of radiation and hydrodynamics. Astrophys. J. 521, 432–450 (1999)

    Article  Google Scholar 

  33. Lions, P.-L.: Mathematical Topics in Fluid Dynamics, vol. 2. Compressible Models. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  34. Mihalas, B., Weibel-Mihalas, B.: Foundations of Radiation Hydrodynamics. Dover, New York (1984)

    MATH  Google Scholar 

  35. Pedregal, P.: Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

  36. Pomraning, G.C.: Radiation Hydrodynamics. Dover, New York (2005)

    Google Scholar 

  37. Ripoll, J.F., Dubroca, B., Duffa, G.: Modelling radiative mean absorption coefficients. Combust. Theory Model. 5, 261–274 (2001)

    Article  MATH  Google Scholar 

  38. Zhong, X., Jiang, J.: Local existence and finite-time blow up in multidimensional radiation hydrodynamics. J. Math. Fluid Mech. 9, 543–564 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work of Šárka Nečasová acknowledges the support of the GAČR (Czech Science Foundation) project P201-13-00522S in the framework of RVO: 67985840.

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Authors and Affiliations

  1. CEA/DAM/DIF, 91297, Arpajon, France

    Bernard Ducomet

  2. Mathematical and Physical Faculty of the Charles University in Prague, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Marek Kobera

  3. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Prague 1, Czech Republic

    Šárka Nečasová

Authors
  1. Bernard Ducomet
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  2. Marek Kobera
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  3. Šárka Nečasová
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Correspondence to Šárka Nečasová.

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Ducomet, B., Kobera, M. & Nečasová, Š. Global Existence of a Weak Solution for a Model in Radiation Magnetohydrodynamics. Acta Appl Math 150, 43–65 (2017). https://doi.org/10.1007/s10440-016-0093-y

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  • DOI: https://doi.org/10.1007/s10440-016-0093-y

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