The homogenization of diffusion-convection equations in non-periodic structures

We consider the homogenization of diffusion-convective problems with given divergence-free velocities in nonperiodic structures defined by sequences of characteristic functions the first sequence . These quence of concentration the second sequence is uniformly bounded in the space of square-summable functions with square-summable derivatives with respect to spatial variables. At the same time, the sequence of time-derivative of product of these concentrations on the characteristic functions, that define a nonperiodic structure, is bounded in the space of square-summable functions from time interval into the conjugated space of functions depending on spatial variables, withsquare-summable derivatives. We prove the strong compactness of the second sequences in the space of quadratically summable functions and use this result to homogenize the corresponding boundary value problems that depend on a small parameter.

Keywords:

Diffusion-convection, homogenization nonperiodic structures, compactness lemma,

PDF

___

[1] Acerbi E, Chiad‘o V, Maso G, Percivale D. An extension theorem from connected sets and homogenization in general periodic domains. Nonlinear Analysis 1992; 5 (5): 481-496. doi: 10.1016/0362-546X(92)90015-7
[2] Adams RA. Sobolev spaces. New York, NY, USA: Academic Press, 1975.
[3] Aubin JP. Un théorème de compacité. Comptes rendus de lÁcadèmie des Sciences 1963; 256 (1): 5042-5044.
[4] Bensoussan A, Lions J, Papanicolau G. Asymptotic Analysis for Periodic Structure. Amsterdam, Netherlands: North Holland Publishing Company, 1978.
[5] Chen X, Jungel A, Liu J. Note on Aubin-Lions-Dubinskii Lemmas. Acta Applicandae Mathematicae 2014; 133 (1): 33-43. doi: 10.1007/s10440-013-9858-8
[6] Jikov VV, Kozlov SM, Oleinik OA. Homogenization of differential operators and integral functionals. Berlin, Germany: Springer-Verlag, 1994.
[7] Kolmogorov AN, Fomin SV. Introductory real analysis. New York, NY, USA: Dover Publications, 1975.
[8] Ladyzhenskaya OA, Solonnikov VA, Uraltseva NN. Linear and Quasilinear Equations of Parabolic Type. Providence, RI, USA: American Mathematical Society, 1968.
[9] Lions JL. Quelques methodes de resolution des problemes aux limites nonlineaire. Paris, France: Dunod, 1969.
[10] Meirmanov A, Zimin R. Compactness result for periodic structures and its application to the homogenization of a diffusion-convection equation. Electronic Journal of Differential Equations 2011; 2011 (115): 1-11.
[11] Meirmanov A, Shmarev S. A compactness lemma of Aubin type and its application to a class of degenerate parabolic equations. Electronic Journal of Differential Equations 2014; 2014 (227): 1-13.
[12] Meirmanov A. Mathematical models for poroelastic flow. Paris, France: Atlantis Press, 2014.
[13] Mikhailov VP, Gushchin AK. Additional chapters of the course “Equations of Mathematical Physics”, Lecture courses REC, Issue 7, V.A. Steklov’s Mathematical Institute. Moscow, Russia: RAS, 2007.
[14] Nguetseng G. A general convergence result for a functional related to the theory of homogenization. SIAM Journal on Mathematical Analysis 1989; 20 (3): 608-623. doi:10.1137/0520043
[15] Rudin W. Principles of mathematical analysis. New York, NY, USA: McGraw-Hill, 1976

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv