  In this paper, we revisit the notion of temporal intermittency to obtain
sharp nonuniqueness results for linear transport equations. We construct
divergence-free vector fields with sharp Sobolev regularity $L^1_t W^{1,p}$ for
all $p<\infty$ in space dimensions $d\geq 2$ whose transport equations admit
nonunique weak solutions belonging to $L^p_tC^k$ for all $p<\infty$ and $k\in
\mathbb{N}$. In particular, our result shows that the time-integrability
assumption in the uniqueness of the DiPerna-Lions theory is sharp. The same
result also holds for transport-diffusion equations with diffusion operators of
arbitrarily large order in any dimensions $d \geq 2$.
