I am looking for examples of $L_1$-bounded sequences of vector fields in $L_1=L_1(\mathbb R^n, \mathbb R^n)$ ($n>1$) that have zero divergence (in the distributional sense) such that no subsequence is convergent in $L_1$.
Such sequences are plentiful because the space of such vector fields is infinite-dimensinonal, so ideally I'd like to see such examples with uniformly bounded (in $L_\infty$) partial derivatives but it is not essential.