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 $\epsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\mathrm{\parallel }U(y)-y{\mathrm{\parallel }}_{H^{N+4}}\ll 1$ and $N>1$. We prove that if $\epsilon \ll {\nu }^{1\mathrm{/}2}$, where $\nu$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\epsilon$-close in $H^1$ to $(e^{t\nu {\mathrm{\partial }}_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t\gg {\nu }^{-1\mathrm{/}3}$ 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 ${\nu }^{1\mathrm{/}2}$ for 2D shear flows close to the Couette flow.