The homogenization of diffusion-convection equations in non-periodic structures

Abstract

We consider the homogenization of diffusion-convective problems with given divergence-free velocities innonperiodic structures defined by sequences of characteristic functions (the first sequence). The sequence of concentration(the second sequence) is uniformly bounded in the space of square-summable functions with square-summable derivativeswith respect to spatial variables. At the same time, the sequence of time-derivative of product of these concentrationson the characteristic functions, that define a nonperiodic structure, is bounded in the space of square-summablefunctions from time interval into the conjugated space of functions depending on spatial variables, with square-summablederivatives. We prove the strong compactness of the second sequences in the space of quadratically summable functionsand use this result to homogenize the corresponding boundary value problems that depend on a small parameter.