Enhanced dissipation and Hörmander's hypoellipticity

Abstract

We examine the phenomenon of enhanced dissipation from the perspective of Hörmander'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 \[ \partial_t f + b(y) \partial_x f - νΔf = 0 \text{ on } \mathbb{T} \times (0,1) \times \mathbb{R}_+ \] 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 $b^{\prime }(y)$ 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.

26 pages