The Sobolev Stability Threshold for 2D Shear Flows Near Couette

We consider the 2D Navier-Stokes equation on T x R, with initial datum that is epsilon-close in H-N to a shear flow (U(y), 0), where parallel to U(y) - y parallel to(HN+4) << 1 and N > 1. We prove that if epsilon >> nu(1/2), where nu denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains epsilon-close in H-1 to (et(nu partial derivative yy)U(y), 0) for all t > 0. Moreover, the solution converges to a decaying shear flow for times t >> 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.