We consider the transport of a passive scalar f along a divergence-free velocity vector field uᵈ on the infinite space Rᵈ. We give a quantitative version of the DiPerna-Lions well-posedness theory for Sobolev vector fields u ∈ Lₜ¹Wₓ1,p when 1<p<∞ by giving a uniform decay rate of the DiPerna-Lions commutator. We recover a slightly more general form of the known exponential bound on the mixing rate by a Sobolev vector field u ∈ Lₜ¹Wₓ1,p when 1<p<∞.
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Huysmans et al. (2024) studied this question.
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