Asymptotic Stability in the Critical Space of 2D Monotone Shear Flow in the Viscous Fluid

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.