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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, thesolution converges to a decaying shear flow for times $t \gg \nu^{-1/3}$ by amixing-enhanced dissipation effect, and experiences a transient growth ofgradients. In particular, this shows that the stability threshold in finiteregularity scales no worse than $\nu^{1/2}$ for 2D shear flows close to theCouette flow.

The Sobolev stability threshold for 2D shear flows near Couette