A quantitative DiPerna-Lions theory and exponential mixing rate for passive scalar transport

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 Lₜ¹Wₓ^1, 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 Lₜ¹Wₓ^1, p when 1<p<.