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.