Homogenization of parabolic equations with contrasting coefficients

G. V. Sandrakov

Izvestiya. Mathematics , Tome 63 (1999) no. 5, pp. 1015-1061.

Résumé

We consider non-stationary diffusion problems in a periodic medium with inclusions filled with a material of small conductivity. We propose homogenized equations whose solutions approximate those of the problems under consideration. We prove estimates for the accuracy of this approximation as the period of the medium and the conductivity coefficient tend to zero. The form of the homogenized equations and the accuracy estimates depend essentially on the asymptotic behaviour of the conductivity coefficient in relation to the square of the period.

Bibliographie

[1] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[2] Bakhvalov N. S., Panasenko G. P., Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR | Zbl

[3] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993 | MR | Zbl

[4] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[5] Sandrakov G. V., Osrednenie linearizovannoi sistemy gidrodinamiki s maloi vyazkostyu i skorost zvuka v smesyakh, Preprint No 178, OVM AN SSSR, M., 1987 | MR

[6] Hornung U., Showalter R. E., “Diffusion models for fractured media”, J. Math. Analysis and Applications, 147:1 (1990), 67–80 | DOI | MR

[7] Sandrakov G. V., “Printsipy osredneniya uravnenii s bystroostsilliruyuschimi koeffitsientami”, Matem. sb., 180:12 (1989), 1643–1690

[8] Panasenko G. P., “Mnogokomponentnoe osrednenie protsessov v silno neodnorodnykh strukturakh”, Matem. sb., 181:1 (1990), 134–142 | MR | Zbl

[9] Bakhvalov N. S., Eglit M. E., “O predelnom povedenii periodicheskikh sred s myagkomodulnymi vklyucheniyami”, ZhVM i MF, 35:6 (1995), 905–917 | MR | Zbl

[10] Khruslov E. Ya., Rost G., “Usrednenie uravneniya diffuzii s koeffitsientami, ne udovletvoryayuschimi usloviyu ravnomernoi elliptichnosti”, UMN, 46:6 (1991), 189–190

[11] Hornung U., Jager W., “Diffusion, convection, adsorption, and reaction of chemicals in porous media”, J. Diff. Equat., 92:2 (1991), 199–225 | DOI | MR | Zbl

[12] Arbogast T, Douglas J., Hornung U., “Derivation of the double porosity model of single phase flow via homogenization theory”, SIAM J. Math. Anal., 21:4 (1990), 823–836 | DOI | MR | Zbl

[13] Sandrakov G. V., “Diffuziya v silno neodnorodnoi periodicheskoi srede”, UMN, 51:5 (1996), 208–209

[14] Temam R., Uravneniya Nave–Stoksa, Mir, M., 1981 | MR | Zbl

[15] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1988 | MR

[16] Agranovich M. S., Vishik M. I., “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, UMN, 19:3 (1964), 53–161 | MR | Zbl

[17] Khille E., Fillips R., Funktsionalnyi analiz i polugruppy, IL, M., 1962

[18] Abramovits M., Stigan I., Spravochnik po spetsialnym funktsiyam s formulami, grafikami i tablitsami, Nauka, M., 1979 | MR

[19] Godunov S. K., Uravneniya matematicheskoi fiziki, Nauka, M., 1979 | MR | Zbl

[20] Trenogin V. A., “Razvitie i primenenie metoda Lyusternika–Vishika”, UMN, 25:4 (1970), 123–156 | MR | Zbl