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

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