Enhanced dissipation and H\"ormander's hypoellipticity

We examine the phenomenon of enhanced dissipation from the perspective of H\"ormander's classical theory of second order hypoelliptic operators [31]. Consider a passive scalar in a shear flow, whose evolution is described by the advection-diffusion equation $$\partial_t f + b(y) \partial_x f - \nu \Delta f = 0 \text{ on } \mathbb{T} \times (0,1) \times \mathbb{R}_+$$ with periodic, Dirichlet, or Neumann conditions in $y$. We demonstrate that decay is enhanced on the timescale $T \sim \nu^{-(N+1)/(N+3)}$, where $N-1$ is the maximal order of vanishing of the derivative $b'(y)$ of the shear profile and $N=0$ for monotone shear flows. In the periodic setting, we recover the known timescale of Bedrossian and Coti Zelati [8]. Our results are new in the presence of boundaries.