Exponential Lower Bounds for the Advection-Diffusion Equation with Shear Flows

In this paper, we prove that the $L^2$ norm of spatial mean-free solutions to the advection–diffusion equation on ${\mathbb{T}}^2$ with shear drifts satisfies an \emph{exponential lower bound} in time. This lower bound shows that diffusion can fundamentally suppress passive-scalar mixing.