Homogenized Models with Memory Effect for Heterogeneous Periodic Media
Анотація
The homogenization of initial boundary value problems for heat conduction equations with asymptotically degenerate rapidly oscillating periodic coefficients are considered. Such problems model thermal processes in heterogeneous periodic media. Homogenized problems (whose solutions determine approximate asymptotics for solutions of the original problems) are presented. Estimates for the accuracy of the asymptotics and relevant convergence theorem are discussed. The homogenized problems have the form of initial boundary value problems for integro-differential equations in convolutions. The presence of convolutions in models for media is called the memory effect. Statements about the solvability and regularity for the problems and the homogenized problems are proved. These results are optimal even in the case of zero convolutions, when the homogenized problems coincide with the classical heat conduction problems.
Ключові слова
Повний текст:
PDF (English)Посилання
M. S. Agranovich, M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Russian Math. Surveys, 19 (3) (1964), 53–157. doi: 10.1070/RM1964v019n03ABEH001149
T. Arbogast, J. Douglas, U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (4) (1990), 823–836.
N. S. Bakhvalov, G. P. Panasenko, Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht, 1989.
M. Benes, I. Pazanin, Homogenization of degenerate coupled transport processes in porous media with memory terms, Math Methods in the Applied Sciences, 42 (18) (2019), 6227–6258. doi: 10.1002/mma.5718
A. Bensoussan, J.-L. Lions, G. Papanicolau, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, New York, Oxford, 1978.
J. I. Diaz, D. Gomez-Castro, T. A. Shaposhnikova, M. N. Zubova, A nonlocal memory strange term arising in the critical scale homogenization of a diffusion equation with dynamic boundary conditions, Electron. J. Differ. Equations, 2019 (77) (2019), 1–13.
J. I. Diaz, D. Gomez-Castro, T. A. Shaposhnikova, M. N. Zubova, Classification of homogenized limits of diffusion problems with spatially dependent reaction over critical-size particles, Applicable Analysis, 98 (1–2) (2019), 232–255. doi:10.1080/00036811.2018.1441997
V. Dudar, V. Semenov, Use of symmetric kernels for convolutional neural networks, Advances in Intelligent Systems and Computing, 836 (2019), 3–10. doi:10.1007/978-3-319-97885-7_1
G. Duvaut, J.-L. Lions, Les Inequations en Mecanique et en Physique, Dunod, Paris, 1972.
M. Gahn, M. Neuss-Radu, I. Pop, Homogenization of a reaction-diffusionadvection problem in an evolving micro-domain and including nonlinear boundary conditions, J. Differential Equations, 289 (2021), 95–127. doi: 10.1016/j.jde.-2021.04.013
S. Garttner, P. Frolkovic, P. Knabner, N. Ray, Efficiency and accuracy of micro-macro models for mineral dissolution, Water Resources Research, 56(8) (2020):e2020WR027585. doi: 10.1029/2020WR027585
S. Garttner, P. Frolkovic, P. Knabner, N. Ray, Efficiency of micro-macro models for reactive two-mineral systems, Multiscale Modeling and Simulation, 20 (1) (2022), 433–461. doi: 10.1137/20M1380648
W. Jager, R. Rannacher, J. Warnatz, Reactive Flows, Diffusion and Transport. From Experiments via Mathematical Modeling to Numerical Simulation and Optimization, Springer-Verlag, Berlin, Heidelberg, 2007.
W. Jager, L. Woukeng, Homogenization of Richards’ equations in multiscale porous media with soft inclusions, J. Differential Equations, 281 (2021), 503–549. doi: 10.1016/j.jde.2021.02.012
M. Kelm, S. Garttner, C. Bringedal, B. Flemisch, P. Knabner, N. Ray, Comparison study of phase-field and level-set method for three-phase systems including two minerals, Comput. Geosciences, 26 (2022), 545–570. doi: 10.1007/s10596-022-10142-w
S. I. Lyashko, V. V. Semenov, Controllability of Linear Distributed Systems in Classes of Generalized Actions, Cybernetics and Systems Analysis, 37 (2001), 13–32. doi: 10.1023/A:1016607831284
A. Mielke, S. Reichelt, M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw. Heterog. Media, 9 (2) (2014), 353–382. doi: 10.3934/nhm.2014.9.353
A. Mielke, S. Reichelt, Traveling fronts in a reaction–diffusion equation with a memory term, J. Dynamics and Differential Equations (2022). doi: 10.1007/s10884-022-10133-6
E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag, New York, 1980.
G. V. Sandrakov, Averaging principles for equations with rapidly oscillating coefficients, Mathematics of the USSR - Sbornik, 68 (2) (1991), 503–553.
G. V. Sandrakov, The homogenization of nonstationary equations with contrast coefficients, Doklady Mathematics, 56 (1) (1997), 586–589.
G. V. Sandrakov, Multiphase homogenized models for diffusion in highly nonhomogeneous media, Doklady Mathematics, 70 (1) (2004), 507–511.
G. V. Sandrakov, Multiphase models of nonstationary diffusion arising from homogenization, Computational Mathematics and Mathematical Physics, 44 (10) (2004), 1741–1756.
G. V. Sandrakov, Multiphase homogenized diffusion models for problems with several parameters, Izvestiya Mathematics, 71 (6) (2007), 1193–1252. doi: 10.1070-/IM2007v071n06ABEH002387
G. V. Sandrakov, A. L. Hulianytskyi, Solvability of homogenized problems with convolutions for weakly porous media, J. Numerical and Applied Mathematics, 2(134) (2020), 59–70. (In Russian) doi: 10.17721/2706-9699.2020.2.04
G. Sandrakov, A. Hulianytskyi, V. Semenov, Modeling of filtration processes in periodic porous media, Modeling, Control and Information Technologies, 5 (2021), 90–93. doi: 10.31713/MCIT.2021.28
V. V. Semenov, Solvability of a parabolic transmission problem with the condition of a generalized proper lumped source, Differential Equations, 41 (6) (2005), 878–886. doi: 10.1007/s10625-005-0227-x
T. A. Shaposhnikova, M. N. Zubova, Homogenization of variational inequality for the Laplace operator with nonlinear constraint on the flow in a domain perforated by arbitrary shaped sets. Critical case, J. Mathematical Sciences, 232 (4) (2018), 573–590. doi: 10.1007/s10958-018-3888-8
M. N. Zubova, T. A. Shaposhnikova, Homogenization of a boundary-value problem in a domain perforated by cavities of arbitrary shape with a general nonlinear boundary condition on their boundaries: The case of critical values of the parameters, J. Mathematical Sciences, 244 (2) (2020), 235–253. doi: 10.1007/s10958-019-04616-z
DOI: http://dx.doi.org/10.15421/142206
Посилання
- Поки немає зовнішніх посилань.
Індексування журналу
Журнал розміщено у наукометричних базах, репозитаріях та пошукових системах:
Адреса редколегії: 49050, Україна, Дніпровський національний університет імені Олеся Гончара, вул. Козакова 18, корп. 14, механіко-математичний факультет, д-р фіз.-мат. наук, проф. Когут П.І.
email: p.kogut@i.ua
Це видання має доступ за ліцензією Creative Commons «Attribution» («Атрибуция») 4.0 Всемирная.