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 $L^\infty_{ t,x}$ 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 $L^\infty_t W^{ 1,\infty}_x$ and exhibit a solution that decays with rate $e^{ -C^{ -1} t^2}$. In 4D, we construct a velocity field in $L^\infty_t C^\infty_x$ and exhibit a solution that decays with *some* superexponential rate.
Categories: math.AP