The Sobolev Stability Threshold for 2D Shear Flows Near Couette

We consider the 2D Navier–Stokes equation on $$\mathbb T \times \mathbb R$$T×R, with initial datum that is $$\varepsilon $$ε-close in $$H^N$$HN to a shear flow (U(y), 0), where $$\Vert U(y) - y\Vert _{H^{N+4}} \ll 1$$‖U(y)-y‖HN+4≪1 and $$N>1$$N>1. We prove that if $$\varepsilon \ll \nu ^{1/2}$$ε≪ν1/2, where $$\nu $$ν denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains $$\varepsilon $$ε-close in $$H^1$$H1 to $$(e^{t \nu \partial _{yy}}U(y),0)$$(etν∂yyU(y),0) for all $$t>0$$t>0. Moreover, the solution converges to a decaying shear flow for times $$t \gg \nu ^{-1/3}$$t≫ν-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 $$\nu ^{1/2}$$ν1/2 for 2D shear flows close to the Couette flow.