Extreme temporal intermittency in the linear Sobolev transport: almost smooth nonunique solutions

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 $L_t^1{W}^{1,p}$ for all $p<\mathrm{\infty }$ in space dimensions $d\ge 2$ whose transport equations admit nonunique weak solutions belonging to $L_t^p{C}^k$ for all $p<\mathrm{\infty }$ 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 $d\ge 2$.