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 (varepsilon )-close in (h^n) to a shear flow (u(y), 0), where (vert u(y) - yvert _{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, the solution converges to a decaying shear flow for times (t gg 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.