Enhanced dissipation and Taylor dispersion by a parallel shear flow in an infinite cylinder with unbounded cross section

In this paper, we investigate the long-time behavior of a passive scalar advected by a parallel shear flow in an infinite cylinder with unbounded cross section, in the regime where the viscosity coefficient satisfies $\nu \ll 1$, and in arbitrary spatial dimension. Under the assumption of an infinite cylinder, that is, $x \in \mathbb{R}$, the corresponding Fourier frequency $k$ (often referred to as the streamwise wave number) also ranges over the whole real line $\mathbb{R}$. In this setting, the enhanced dissipation phenomenon only occurs for high frequencies $\nu \le |k|$, whereas for low frequencies $|k| \le \nu$ only the decay of Taylor dispersion appears. It is worth noting that, in the case where $x \in \mathbb{T}$, the Fourier frequency does not contain low frequencies near zero, and thus enhanced dissipation occurs for all nonzero modes. Previously, Coti Zelati and Gallay study in [8] the case of infinite cylinders with bounded cross sections. For the unbounded case considered here, we find that a non-degeneracy condition at infinity is also required.