For advection-diffusion on T^d there exist incompressible velocity fields and initial data whose L2 mass decays double exponentially in 2D, like e^{-Ct^2} in 3D, and superexponentially in 4D.
On the norm equivalence of L yapunov exponents for regularizing linear evolution equations
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Superexponential dissipation enhancement on $\mathbb{T}^d$
For advection-diffusion on T^d there exist incompressible velocity fields and initial data whose L2 mass decays double exponentially in 2D, like e^{-Ct^2} in 3D, and superexponentially in 4D.