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 [33]. Consider a passive scalar in a shear flow, whose evolution is described by the advection–diffusion equation∂tf+b(y)∂xf−νΔf=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.