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

Abstract

We prove the strong compactness of the sequence ${c^{varepsilon}(mathbf{x},t)}$ in $L_2(Omega_T)$, $Omega_T={(mathbf{x},t):mathbf{x}inOmega subset mathbb{R}^3, tin(0,T)}$, bounded in $W^{1,0}_2(Omega_T)$ with the sequence of time derivative ${partial/partial tig(chi(mathbf{x}/varepsilon) c^{varepsilon}ig)}$ bounded in the space $L_2ig((0,T); W^{-1}_2(Omega)ig)$. 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 $L_2(Omega_T)$.

Keywords

• Weak, strong and two-scale convergence
• homogenization
• diffusion-convection