Superexponential dissipation enhancement on ${\mathbb{T}}^d$

We construct incompressible velocity fields that exhibit faster than exponential dissipation for particular solutions to the advection-diffusion equation on ${\mathbb{T}}^d$. In 2D, we construct a velocity field in $L_{t,x}^{\mathrm{\infty }}$ and exhibit a solution that decays with double exponential rate $e^{-C^{-1}{e}^{C^{-1}t}}$. In 3D, we construct a velocity field in $L_t^{\mathrm{\infty }}{W}_x^{1,\mathrm{\infty }}$ and exhibit a solution that decays with rate $e^{-C^{-1}{t}^2}$. In 4D, we construct a velocity field in $L_t^{\mathrm{\infty }}{C}_x^{\mathrm{\infty }}$ and exhibit a solution that decays with some superexponential rate.