Lack of compactness in two-scale convergence

This article deals with the links between compensated compactness and two-scale convergence. More precisely, we ask the following question: Is the div-curl compactness assumption sufficient to pass to the limit in a product of two sequences which two-scale converge with respect to the pair of variables (x, x/ε)? We reply in the negative. Indeed, the div-curl assumption allows us to control oscillations which are faster than 1/ε but not the slower ones.