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 ν 1 2 for perturbations in the critical space H x log L y 2 . Specifically, if the initial velocity V in and the corresponding vorticity W in are ν 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 - ( - ∂ y b in ) ‖ H x log L y 2 ≤ ε ν 1 2 , then the velocity V ( t ) stay ν 1 2 -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 ≠ ‖ L 2 ≲ ε ν 1 2 e - c ν 1 3 t and ‖ V ≠ ‖ L t 2 L x , y 2 ≲ ε ν 1 2 . In the proof, we construct a time-dependent wave operator corresponding to the Rayleigh operator b ( t , y ) Id - ∂ yy b ( t , y ) Δ - 1 , which could be useful in future studies.