Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 115, pp. 1-11.

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

Authors: Anvarbek Meirmanov (Belgorod State Univ., Russia)
         Reshat Zimin       (Belgorod State Univ., Russia)

Abstract:
 We prove the strong compactness of the sequence
 $\{c^{\varepsilon}(\mathbf{x},t)\}$ in $L_2(\Omega_T)$,
 $\Omega_T=\{(\mathbf{x},t):\mathbf{x}\in\Omega
 \subset  \mathbb{R}^3, t\in(0,T)\}$, bounded in
 $W^{1,0}_2(\Omega_T)$ with the sequence of time derivative
 $\{\partial/\partial t\big(\chi(\mathbf{x}/\varepsilon)
 c^{\varepsilon}\big)\}$  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
 $\{\mathbf{v}^{\varepsilon}(\mathbf{x},t)\}$
 weakly convergent in $L_2(\Omega_T)$.

Submitted April 10, 2011. Published September 06, 2011.
Math Subject Classifications: 35B27, 46E35, 76R99.
Key Words: Weak, strong and two-scale convergence; homogenization;
           diffusion-convection.