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., $\|V_{in}-(b_{in}(y),0)\|_{L_{x,y}^2}+\|W_{in}-(-\partial_yb_{in})\|_{H^{log}_xL^2_y}\leq \epsilon \nu^{\frac{1}{2}}$, 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 $\|W_{\neq}\|_{L^2}\lesssim \epsilon\nu^{\frac{1}{2}}e^{-c\nu^{\frac{1}{3}}t}$ and $\|V_{\neq}\|_{L^2_tL^2_{x,y}}\lesssim \epsilon\nu^{\frac{1}{2}}$. In the proof, we construct a time-dependent wave operator corresponding to the Rayleigh operator $b(t,y)\partial_x-\partial_{yy}b(t,y)\partial_x\Delta^{-1}$, which could be useful in future studies.