Stability threshold of nearly-Couette shear flows with Navier boundary conditions in 2D

In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, $\mathbb{T}\times [-1,1]$, supplemented with Navier boundary conditions $\omega {\mathrm{\mid }}_{y=\pm 1}=0$. Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of $\nu$. On the other hand, the nonzero modes are assumed size $O({\nu }^{\frac{1}{2}})$ in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the $\mathit{\text{inviscid}}\mathit{\text{damping}}$, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework.