Pseudo-Gevrey smoothing for the passive scalar equations near Couette

Abstract

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 $T\times [-1,1]$ 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-ν regularity is with respect to ${\partial }_x$ and a time-dependent adapted vector-field Γ 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, [5], 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.