Electron. J. Diff. Equ., Vol. 2011 (2011), No. 115, pp. 1-11.

Compactness result for periodic structures and its application to the homogenization of a diffusion-convection equation

Anvarbek Meirmanov, Reshat Zimin

Abstract:
We prove the strong compactness of the sequence $\{c^{\varepsilon}({\bf x},t)\}$ in $L_2(\Omega_T)$, $\Omega_T=\{({\bf x},t):{\bf x}\in\Omega \subset \mathbb{R}^3, t\in(0,T)\}$, bounded in $W^{1,0}_2(\Omega_T)$ with the sequence of time derivative $\{\partial/\partial t\big(\chi({\bf x}/\varepsilon) c^{\varepsilon}\big)\}$ bounded in the space $L_2\big((0,T); W^{-1}_2(\Omega)\big)$. As an application we consider the homogenization of a diffusion-convection equation with a sequence of divergence-free velocities $\{{\bf v}^{\varepsilon}({\bf x},t)\}$ weakly convergent in $L_2(\Omega_T)$.

Submitted April 10, 2011. Published September 6, 2011.
Math Subject Classifications: 35B27, 46E35, 76R99.
Key Words: Weak, strong and two-scale convergence; homogenization; diffusion-convection.

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Anvarbek M. Meirmanov
Department of mahtematics, Belgorod State University
ul.Pobedi 85, 308015 Belgorod, Russia
email: meirmanov@bsu.edu.ru
Reshat Zimin
Department of mahtematics, Belgorod State University
ul.Pobedi 85, 308015 Belgorod, Russia
email: reshat85@mail.ru

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