Solvability of equations with convolutions that arise in homogenization problems
DOI:
https://doi.org/10.15407/dopovidi2021.06.015Keywords:
initial-boundary-value problem, homogenized equation, a priori estimate, Laplace transformAbstract
We consider the initial-boundary-value problems for the non-stationary equations of filtration in porous media. Such problems are relevant in the underground water pollution control. We consider the periodic media with a small microscale coefficient as models of porous media. We present the solvability and regularity theorems for the corresponding homogenized problems with convolutions. These theorems are formulated for general input data and non-homogeneous initial conditions, and they extend the classical solvability theorems for the heat equation. To prove the theorems, we use the a priori estimate method and the well-known Agranovich—Vishik method.
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Sandrakov, G. V. (1997). The homogenization of nonstationary equations with contrast coefficients. Dokl. Mathematics, 56, No.1, рр. 586-589.
Sandrakov, G. V. (1999). Homogenization of parabolic equations with contrasting coefficients. Izvestiya: Math., 63, No. 5, pp. 1015-1061.
Sandrakov, G. V. (2007). Multiphase homogenized diffusion models for problems with several parameters. Izvestiya: Mathematics, 71, No. 6, рр. 1193-1252.
Duvaut, G. & Lions, J.-L. (1972). Les inequations en mecanique et en physique. Dunod, Paris.
Jager, W., Rannacher, R. & Warnatz, J. (Eds.) (2007). Reactive Flows, Diffusion and Transport. From Experiments via Mathematical Modeling to Numerical Simulation and Optimization. Berlin, Heidelberg: Springer.
Amosov, A. A. & Zlotnik, A. A. (1998). On the quasi-averaging of a system of equations of the one-dimensional motion of a viscous heat-conducting gas with rapidly oscillating data. Comput. Math. Math. Phys., 38, No. 7, рр. 1152-1167.
Amosov, A. A. & Zlotnik, A. A. (2001). Justification of two-scale averaging of equations of one-dimensional nonlinear thermoviscoelasticity with nonsmooth data. Comput. Math. Math. Phys., 41, No. 11, рр. 1630-1650.
Agranovich, M. S. & Vishik, M. I. (1964). Elliptic problems with a parameter and parabolic problems of general type. Russian Math. Surveys, 19, No. 3, рр. 53-157.
Sandrakov, G.V. & Hulianytskyi, A. L. (2020). Solvability of homogenized problems with convoluitions for weakly porous media. J. Numer. Appl. Mathematics, № 2 (134), рр. 59-70 (in Russian).
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