In this paper, we study the long-time behavior of the solutions to the two-dimensional incompressible free Navier Stokes equation (without forcing) with small viscosity \(\nu \) , when the initial data is close to stable monotone shear flows. We prove the asymptotic stability and obtain the sharp stability threshold \(\nu ^{\frac{1}{2}}\) for perturbations in the critical space \(H^{log}_xL^2_y\) . Specifically, if the initial velocity \(V_{in}\) and the corresponding vorticity \(W_{in}\) are \(\nu ^{\frac{1}{2}}\) -close to the shear flow \((b_{in}(y),0)\) in the critical space, i.e., \(\Vert V_{in}-(b_{in}(y),0)\Vert _{L_{x,y}^2}+\Vert W_{in}-(-\partial _yb_{in})\Vert _{H^{log}_xL^2_y}\le \varepsilon \nu ^{\frac{1}{2}}\) , then the velocity V(t) stay \(\nu ^{\frac{1}{2}}\) -close to a shear flow (b(t, y), 0) that solves the free heat equation \((\partial _t-\nu \partial _{yy})b(t,y)=0\) . We also prove the enhanced dissipation and inviscid damping, namely, the nonzero modes of vorticity and velocity decay in the following sense \(\Vert W_{\ne }\Vert _{L^2}\lesssim \varepsilon \nu ^{\frac{1}{2}}e^{-c\nu ^{\frac{1}{3}}t}\) and \(\Vert V_{\ne }\Vert _{L^2_tL^2_{x,y}}\lesssim \varepsilon \nu ^{\frac{1}{2}}\) . In the proof, we construct a time-dependent wave operator corresponding to the Rayleigh operator \(b(t,y)\textrm{Id}-\partial _{yy}b(t,y)\Delta ^{-1}\) , which could be useful in future studies.
