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

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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 < ∞ in space dimensions d ≥ 2 whose transport equations admit nonunique weak solutions belonging to L p t C k for all p < ∞ and k ∈ ‫.ގ‬ In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna-Lions theory is essential. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions d ≥ 2.