Enhanced dissipation and Hörmander's hypoellipticity.

We examine the phenomenon of enhanced dissipation from the perspective of Hörmander's classical theory of second order hypoelliptic operators [33]. Consider a passive scalar in a shear flow, whose evolution is described by the advection–diffusion equation ∂ t f + b (y) ∂ x f − ν Δ f = 0 on T × (0 , 1) × R + with periodic, Dirichlet, or Neumann conditions in y. We demonstrate that decay is enhanced on the timescale T ∼ ν − (N + 1) / (N + 3) , where N 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. [ABSTRACT FROM AUTHOR]