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 ν, when the initial data is close to stable monotone shear flows. We prove the asymptotic stability and obtain the sharp stability threshold ν¹² for perturbations in the critical space H_x^logL_y². Specifically, if the initial velocity V_in and the corresponding vorticity W_in are ν¹²-close to the shear flow (b_in(y),0) in the critical space, i.e., ‖V_in-(b_in(y),0)‖L_x,y2+‖Win-(-∂ybin)‖_HxlogLy2≤εν¹², then the velocity V(t) stay ν¹²-close to a shear flow (b(t, y), 0) that solves the free heat equation (∂_t-ν∂_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_≠‖L₂≲εν¹²e^-cν13t and ‖V_≠‖_Lt2Lx,y2≲εν¹². In the proof, we construct a time-dependent wave operator corresponding to the Rayleigh operator b(t,y)Id-∂_yyb(t,y)Δ^-1, which could be useful in future studies.