We give an example of a weakly mixing \({{\,\textrm{BV}\,}}\) divergence-free vector field \(b\in L^\infty ([0,1],{{\,\textrm{BV}\,}}(\mathbb {T}^2))\) which is not strongly mixing, in the setting introduced in Bianchini and Zizza (Comm Math Phys 402:1953-2009, 2023), Elgindi and Zlatoš (Adv Math 356, 2019). This is an example of a deterministic vector field with a weakly mixing but not strongly mixing behaviour, while previously were known only examples of strongly mixing vector fields. The construction here is based on a work of Chacon (Proceedings of the American Mathematical Society 22:59-562, 1969) who developed a general framework to generate weakly mixing properties. In particular, he constructed a weakly mixing automorphism (a measure-preserving invertible map) which is not strongly mixing on \(([0,1],\mathcal {B}([0,1]),|\cdot |)\) , where \(\mathcal {B}([0,1])\) are the Borel subsets of [0, 1] and \(|\cdot |\) is the one-dimensional Lebesgue measure. The idea of the proof is to decompose the motion into simple movements of subquares that can be obtained by composition of divergence-free vector fields. Here the example is given in the two-dimensional torus, after a short revision of the one-dimensional setting, but the construction can be extended to any dimension.
