Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)32574ArticlesСтатьиResearch ArticleNonstationary Problem of Complex Heat Transfer in a System of Semitransparent Bodies with Radiation Di use Re ection and Refraction Boundary-Value ConditionsНестационарная задача сложного теплообмена в системе полупрозрачных тел с краевыми условиями диффузного отражения и преломления излученияAmosovA. A.АмосовА. А.amosovaa@mpei.ruNational Research University “Moscow Power Engineering Institute”Национальный исследовательский университет «Московский энергетический институт»1512201659VOL 59, NO (2016)ТОМ 59, № (2016)53414112022Copyright ©; 2022, Contemporary Mathematics. Fundamental DirectionsCopyright ©; 2022, Современная математика. Фундаментальные направления2022Contemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направленияhttps://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/CMFD/article/view/32574We consider a nonstationary initial-boundary value problem describing complex (radiative-con-ductive) heat transfer in a system of semitransparent bodies. To describe radiation propagation, we use the transport equation with radiation di use re ection and refraction boundary-value conditions. We take into account that the radiation intensity and optical properties of bodies depend on the radiation frequency. The unique solvability of a weak solution is established. The comparison theorem is proved. A priori estimates of a weak solution are obtained as well as its regularity.Рассматривается нестационарная начально-краевая задача, описывающая сложный (радиационно-кондуктивный) теплообмен в системе полупрозрачных тел. Для описания распространения излучения используется уравнение переноса излучения с краевыми условиями диффузного отражения и диффузного преломления излучения. Учтена зависимость интенсивности излучения и оптических свойств тел от частоты излучения. Установлены существование и единственность слабого решения. Доказана теорема сравнения. Выведены априорные оценки слабого решения и получен результат о его регулярности.1.Гренкин Г. В., Чеботарев А. Ю. Нестационарная задача сложного теплообмена// Журн. выч. мат. и мат. физ. - 2014. - 54, № 11. - С. 1806-1816.2.Ковтанюк А. Е., Чеботарев А. Ю. Стационарная задача сложного теплообмена// Журн. выч. мат. и мат. физ. - 2014. - 54, № 4. - С. 711-719.3.Ковтанюк А. Е., Чеботарев А. Ю. Стационарная задача свободной конвекции с радиационным теплообменом// Дифф. уравн. - 2014. - 50, № 12. - С. 1590.4.Ладыженская О. А., Солонников В. А., Уральцева Н. Н. Линейные и квазилинейные уравнения параболического типа. - М.: Наука, 1967.5.Михайлов В. П. Дифференциальные уравнения в частных производных. - М.: Наука, 1976.6.Agoshkov V. I. Boundary value problems for transport equations: functional spaces, variational statements, regularity of solutions. - Boston-Basel-Berlin: Birkhauser, 1998.7.Amosov A. A. The solvability of a problem of radiation heat transfer// Soviet Phys. Dokl. - 1979. - 24, № 4. - С. 261-2628.Amosov A. A. The limit connection between two problems of radiation heat transfer// Soviet Phys. Dokl. - 1979. - 24, № 6. - С. 439-4419.Amosov A. A. Stationary nonlinear nonlocal problem of radiative-conductive heat transfer in a system of opaque bodies with properties depending on radiation frequency//j. Math. Sci. (N. Y.) - 2010. - 164, № 3. - С. 309-34410.Amosov A. A. Nonstationary nonlinear nonlocal problem of radiative-conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency//j. Math. Sci. (N. Y.) - 2010. - 165, № 1. - С. 1-4111.Amosov A. A. Boundary value problem for the radiation transfer equation with re ection and refraction conditions// J. Math. Sci. (N. Y.) - 2013. - 191, № 2. - С. 101-149.12.Amosov A. A. Boundary value problem for the radiation transfer equation with di use re ection and refraction conditions// J. Math. Sci. (N. Y.) - 2013. - 193, № 2. - С. 151-176.13.Amosov A. A. Some properties of boundary value problem for radiative transfer equation with di use re ection and refraction conditions// J. Math. Sci. (N. Y.) - 2015. - 207, № 2. - С. 118-141.14.Cessenat M. The´ore`mes de trace Lp pour des espaces de fonctions de la neutronique// C. R. Acad. Sci., Paris Se´r. I - 1984. - 299. - С. 831-834.15.Cessenat M. The´ore`mes de trace pour des espaces de fonctions de la neutronique// C. R. Acad. Sci. Paris Se´r. I. - 1985. - 300.- С. 89-92.16.Dautray R., Lions J.-L. Mathematical analysis and numerical methods for science and technology. Vol. 6: Evolution problems II. - Berlin: Springer, 2000.17.Druet P.-E. Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in Lp (p ?: 1)// Math. Methods Appl. Sci. - 2009. - 32, № 32. - С. 135-166.18.Druet P.-E. Existence for the stationary MHD equations coupled to heat transfer with nonlocal radiation e ects// Czechoslovak Math. J. - 2009. - 59. - С. 791-82519.Druet P.-E. Existence of weak solution to time-dependent MHD equations coupled to heat transfer with nonlocal radiation boundary conditions// Nonlinear Anal. Real World Appl. - 2009. - 10. - С. 2914-2936.20.Druet P.-E. Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side// Appl. Math. - 2010. - 55, № 2. - С. 111-149.21.Gilbarg D., Trudinger N. Elliptic partial di erential equations of second order. - Berlin: Springer, 1983.22.Kelley C. T. Existence and uniqueness of solutions of nonlinear systems of conductive-radiative heat transfer equations// Transport Theory Statist. Phys. - 1996. - 25, № 2. - С. 249-260.23.Kovtanyuk A. E., Chebotarev A. Yu., Botkin N. D. Unique solvability of a steady-state complex heat transfer model// Commun. Nonlinear Sci. Numer. Simul. - 2015. - 20, № 3. - С. 776-784.24.Kovtanyuk A. E., Chebotarev A. Yu., Botkin N. D., Ho mann K.-H. The unique solvability of a complex 3D heat transfer problem// J. Math. Anal. Appl. - 2014. - 409. - С. 808-815.25.Krˇizˇek M., Liu L. On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type// Appl. Math. - 1996. - 24, № 1. - С. 97-107.26.Laitinen M. T. Asymptotic analysis of conductive-radiative heat transfer// Asymptot. Anal. - 2002. - 29.- С. 323-342.27.Laitinen M. T., Tiihonen T. Integro-di erential equation modelling heat transfer in conducting, radiating and semitransparent materials// Math. Methods Appl. Sci. - 1998. - 21. - С. 375-392.28.Laitinen M., Tiihonen T. Conductive-radiative heat transfer in grey materials// Quart. Appl. Math. - 2001. - 59. - С. 737-768.29.Modest F. M. Radiative heat transfer. - Amsterdam, etc.: Academic Press, 2003.30.Necati O¨ zis¸ik M. Radiative transfer and interactions with conduction and convection. - New York, etc.: Willey & Sons, 1973.31.Pinnau R. Analysis of optimal boundary control for radiative heat transfer modelled by SP1 system// Commun. Math. Sci. - 2007. - 5, № 4. - С. 951-969.32.Siegel R., Howell J. R. Thermal radiation heat transfer. - New York-London: CRC Press, 2001.33.Sparrow E. M., Cess R. D. Radiation heat transfer. - New York: Hemisphere Pub. Corp., 1978.