A compactness result for non-periodic structures and its application to homogenization of diffusion-convection equations

The paper proves the strong compactness of the sequence $\{\tilde{c}^{ \varepsilon}(\boldsymbol{x},t)\}$ in $\mathbb{L}_{2}(\Omega_{T})$, $\Omega_{T}=\Omega\times(0,\\T)$, $\Omega\subset \mathbb{R}^{3}$, bounded in the space $\mathbb{W}^{1,0}_{2}(\Omega_{T})$ with the sequence of time derivatives $\Big\{ \displaystyle \frac{\partial}{\partial t}\big(\chi(\boldsymbol{x},t,\frac{\boldsymbol{x}}{\varepsilon})\Big.$ $\Big.\tilde{c}^{ \varepsilon}(\boldsymbol{x},t)\big) \Big\}$ bounded in the space $\mathbb{L}_{2}\big((0,T);\mathbb{W}^{-1}_{2}(\Omega)\big)$, where characteristic function $\chi(\boldsymbol{x},t,\boldsymbol{y})$ is $1$-periodic in a variable $\displaystyle \boldsymbol{y}\in Y= \left(-\frac{1}{2},\frac{1}{2} \right)^{3}\subset \mathbb{R}^{3}$.
As an application we consider the homogenization of a diffusion-convection equation in non-periodic structure, given by $1$-periodic in $\boldsymbol{y}$ characteristic function $\chi(\boldsymbol{x},t,\boldsymbol{y})$ with a sequence of divergent-free velocities $\{\boldsymbol{v}^{\varepsilon}(\boldsymbol{x},t)\}$ weakly convergent in $\mathbb{L}_{2}(\Omega_{T})$.