  We construct a large class of incompressible vector fields with Sobolev
regularity, in dimension $d \geq 3$, for which the chain rule problem has a
negative answer. In particular, for any renormalization map $\beta$ (satisfying
suitable assumptions) and any (distributional) renormalization defect $T$ of
the form $T = {\rm div}\, h$, where $h$ is an $L^1$ vector field, we can
construct an incompressible Sobolev vector field $u \in W^{1, \tilde p}$ and a
density $\rho \in L^p$ for which ${\rm div}\, (\rho u) =0$ but ${\rm div}\,
(\beta(\rho) u) = T$, provided $1/p + 1/\tilde p \geq 1 + 1/(d-1)$
