  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.
