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

Abstract

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 $L_{t,x}^{\infty }$ 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 $L_t^{\infty }{W}_x^{1,\infty }$ and exhibit a solution that decays with rate $e^{-C^{-1}{t}^2}$. In 4D, we construct a velocity field in $L_t^{\infty }{C}_x^{\infty }$ and exhibit a solution that decays with *some* superexponential rate.

32 pages