Monotone polygon

Pure mathematics

Mathematical physics

Boundary value problem

Linear differential equation

Dissipation

Thermodynamics

Physics

Scalar (mathematics)

Mathematics

Mathematical analysis

Geometry

Method of characteristics

Dirichlet distribution

Partial differential equation

Hypoelliptic operator

Journal of Functional Analysis

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.

Enhanced dissipation and Hörmander's hypoellipticity