Abstract
We consider a mathematical model describing a nonstationary Stokes flow in a fine-dispersion mixture of viscous incompressible fluids with rapidly oscillating initial data. We perform homogenization of the model as the frequency of oscillations tends to infinity; this leads to the problem of finding effective coefficients of the averaged equations. To solve this problem, we propose and implement a method which bases on supplementing the averaged system with the Cauchy problem for the kinetic Tartar equation whose unique solution is the Tartar H-measure. Thereby we construct a correct closed model for describing the motion of a homogeneous mixture, because the effective coefficients of the averaged equations are uniquely expressed in terms of the H-measure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Antontsev S. N., Kazhikhov A. V., and Monakhov V. N., Boundary Value Problems of Inhomogeneous Fluid Mechanics [in Russian], Nauka, Novosibirsk (1983).
Temam R., Navier-Stokes Equations. Theory and Numerical Analysis [Russian translation], Mir, Moscow (1981).
Zhikov V. V., Kozlov S. M., and Olecnik O. A., Averaging of Differential Operators [in Russian], Fiz. Mat. Lit., Moscow (1993).
Bakhvalov N. S. and Panasenko G. P., Averaging Processes in Periodic Media [in Russian], Nauka, Moscow (1984).
Sanchez-Palencia E., Non-Homogeneous Media and Vibration Theory [Russian translation], Mir, Moscow (1984).
Allaire G., “Homogenization and two-scale convergence,” SIAM J. Math. Anal., 23, No. 6, 1482–1518 (1992).
Tartar L., “H-measures, a new approach for studying homogenisation oscillations and concentration effects in partial differential equations,” Proc. Roy. Soc. Edinburgh Sect. A, 115, No. 3/4, 193–230 (1990).
DiPerna R. J. and Lions P. L., “Ordinary differential equations, transport theory and Sobolev spaces,” Invent. Math., 98, 511–547 (1989).
Gérard P., “Microlocal defect measures,” Comm. Partial Differential Equations, 16, 1761–1794 (1991).
Sazhenkov S. A., “Solutions of the problem of motion of a viscous incompressible fluid with rapidly oscillating initial data,” Dinamika Sploshn. Sredy (Novosibirsk), No. 113, 123–134 (1998).
Bourbaki N., Integration: Measures. Integration of Measures [Russian translation], Nauka, Moscow (1967).
Málek J., Nečas J., Rokyta M., and Ružička M., Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman and Hall, London (1996).
Sazhenkov S. A., “Cauchy problem for Tartar equation,” Proc. Roy. Soc. Edinburgh Sect. A (to appear).
Sazhenkov S. A., “Generalized Lagrange coordinates in the case of a nonsmooth solenoidal velocity field,” Dinamika Sploshn. Sredy (Novosibirsk), No. 114, 74–77 (1999).
Ladyzhenskaya O. A., Mathematical Problems of the Dynamics of a Viscous Incompressible Fluid [in Russian], Nauka, Moscow (1970).
Sazhenkov S. A., “On averaging of multidimensional parabolic differential operators in hydrodynamics,” Izv. Altačsk. Univ., Barnaul, 2001, No. 1, pp. 43–47. (On-line ISSN 1561–9451, http://tbs.dcn-asu.ru/news/.)
Zhikov V. V., Kozlov S. M., and Oleľnik O. A., “On G-convergence of parabolic operators,” Uspekhi Mat. Nauk, 36, No. 1, 11–58 (1981).
Reshetnyak Yu. G., Stability Theorems in Geometry and Analysis [in Russian], Nauka, Novosibirsk (1982).
Murat F., “H-convergence,” in: Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, 1978, Vol. 77–78, p. 34.
Stein E. M., Singular Integrals and Differentiability Properties of Functions [Russian translation], Mir, Moscow (1973).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sazhenkov, S.A. The Tartar Equation for Homogenization of a Model of the Dynamics of Fine-Dispersion Mixtures. Siberian Mathematical Journal 42, 1142–1155 (2001). https://doi.org/10.1023/A:1012804929542
Issue Date:
DOI: https://doi.org/10.1023/A:1012804929542