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On the well-posedness of the Cauchy problem for the equation of radiative transfer with Fresnel matching conditions

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Abstract

We study the well-posedness of the Cauchy problem for the nonstationary equation of radiative transfer in a three-dimensional bounded domain with Fresnel matching conditions on the interfaces. We prove the existence of a unique strongly continuous semigroup of resolvent operators, and obtain stabilization conditions for nonstationary solutions.

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References

  1. Vladimirov V. S., “Mathematical problems of the one-velocity theory of transport of particles,” Trudy Mat. Inst. Akad. Nauk SSSR, 61, 3–158 (1961).

    Google Scholar 

  2. Cercignani C., Theory and Applications of the Boltzmann Equation [Russian translation], Mir, Moscow (1978).

    Google Scholar 

  3. Novikov V. M. and Shikhov S. B., The Theory of Parameter Action on Neutron Transport [in Russian], Energoizdat, Moscow (1982).

    Google Scholar 

  4. Voigt J., “Positivity in time dependent linear transport theory,” Acta Appl. Math., 2, 311–331 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  5. Germogenova T. A., Local Properties for Solutions of the Transport Equation [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  6. Maslova N. B., “Mathematical methods for studying the Boltzmann equation,” St. Petersburg Math. J., 3, No. 1, 1–43 (1992).

    MathSciNet  Google Scholar 

  7. Anikonov D. S. and Konovalova D. S., “The kinetic transport equation in the case of Compton scattering,” Siberian Math. J., 43, No. 5, 795–807 (2002).

    Article  MathSciNet  Google Scholar 

  8. Anikonov D. S. and Konovalova D. S., “The boundary-value problem for the transport equation with purely Compton scattering,” Siberian Math. J., 46, No. 1, 1–12 (2005).

    Article  MathSciNet  Google Scholar 

  9. Prokhorov I. V., “Boundary value problem of radiation transfer in an inhomogeneous medium with reflection conditions on the boundary,” Differential Equations, 36, No. 6, 943–948 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  10. Prokhorov I. V., “On the solubility of the boundary-value problem of radiation transport theory with generalized conjugation conditions on the interfaces,” Izv. Math., 67, No. 6, 1243–1266 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  11. Prokhorov I. V., “On the structure of the continuity set of the solution to a boundary-value problem for the radiation transfer equation,” Math. Notes, 86, No. 2, 234–248 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  12. Kovtanyuk A. E. and Prokhorov I. V., “A boundary-value problem for the polarized-radiation transfer equation with Fresnel interface conditions for a layered medium,” J. Comput. Appl. Math., 235, No. 8, 2006–2014 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  13. Prokhorov I. V., “Solvability of the initial-boundary value problem for an integrodifferential equation,” Siberian Math. J., 53, No. 2, 301–309 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  14. Prokhorov I. V., “The Cauchy problem for the radiative transfer equation with generalized conjugation conditions,” Comp. Math. Math. Phys., 53, No. 5, 588–600 (2013).

    Article  MathSciNet  Google Scholar 

  15. Amosov A. A., “Boundary value problems for the radiation transfer equation with reflection and refraction conditions,” Vestnik MÉI, No. 1, 99–108 (2014).

    Google Scholar 

  16. Amosov A. A., “Boundary value problem for the radiation transfer equation with diffuse reflection and refraction conditions,” J. Math. Sci., 193, No. 2, 151–176 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  17. Mizohata S., The Theory of Partial Differential Equations, Cambridge University Press, London (1973).

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Institute of Applied Mathematics, Far Eastern Branch of the Russian Academy of Sciences, Vladivostok, Russia

    I. V. Prokhorov & A. A. Sushchenko

  2. Far Eastern Federal University, Vladivostok, Russia

    I. V. Prokhorov & A. A. Sushchenko

Authors
  1. I. V. Prokhorov
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  2. A. A. Sushchenko
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Corresponding author

Correspondence to I. V. Prokhorov.

Additional information

Original Russian Text Copyright © 2015 Prokhorov I.V. and Sushchenko A.A.

The authors were supported by the Russian Science Foundation (Grant 14–11–00079).

Vladivostok. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 4, pp. 922–933, July–August, 2015; DOI: 10.17377/smzh.2015.56.415.

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Prokhorov, I.V., Sushchenko, A.A. On the well-posedness of the Cauchy problem for the equation of radiative transfer with Fresnel matching conditions. Sib Math J 56, 736–745 (2015). https://doi.org/10.1134/S0037446615040151

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  • DOI: https://doi.org/10.1134/S0037446615040151

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