The Sobolev stability threshold for 2D shear flows near Couette

We consider the 2D Navier-Stokes equation on $\mathbb T \times \mathbb R$, with initial datum that is $\varepsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\| U(y) - y\|_{H^{N+4}} \ll 1$ and $N>1$. We prove that if $\varepsilon \ll \nu^{1/2}$, where $\nu$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\varepsilon$-close in $H^1$ to $(e^{t \nu \partial_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t \gg \nu^{-1/3}$ by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than $\nu^{1/2}$ for 2D shear flows close to the Couette flow.