Pseudo-Gevrey Smoothing for the Passive Scalar Equations near Couette

In this article, we study the regularity theory for two linear equations that are important in fluid dynamics: the passive scalar equation for (time-varying) shear flows close to Couette in $\mathbb T \times [-1,1]$ with vanishing diffusivity $\nu \to 0$ and the Poisson equation with right-hand side behaving in similar function spaces to such a passive scalar. The primary motivation for this work is to develop some of the main technical tools required for our treatment of the (nonlinear) 2D Navier-Stokes equations, carried out in our companion work. Both equations are studied with homogeneous Dirichlet conditions (the analogue of a Navier slip-type boundary condition) and the initial condition is taken to be compactly supported away from the walls. We develop smoothing estimates with the following three features: [1] Uniform-in-$\nu$ regularity is with respect to $\partial_x$ and a time-dependent adapted vector-field $\Gamma$ which approximately commutes with the passive scalar equation (as opposed to `flat' derivatives), and a scaled gradient $\sqrt{\nu} \nabla$; [2] $(\partial_x, \Gamma)$-regularity estimates are performed in Gevrey spaces with regularity that depends on the spatial coordinate, $y$ (what we refer to as `pseudo-Gevrey'); [3] The regularity of these pseudo-Gevrey spaces degenerates to finite regularity near the center of the channel and hence standard Gevrey product rules and other amenable properties do not hold. Nonlinear analysis in such a delicate functional setting is one of the key ingredients to our companion paper, \cite{BHIW24a}, which proves the full nonlinear asymptotic stability of the Couette flow with slip boundary conditions. The present article introduces new estimates for the associated linear problems in these degenerate pseudo-Gevrey spaces, which is of independent interest.