The Sobolev Stability Threshold for 2D Shear Flows Near Couette

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