Asymptotic stability in the critical space of 2D monotone shear flow in the viscous fluid

Abstract

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 $H_x^{log}{L}_y^2$. 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., $\Vert V_{in}-(b_{in}(y),0){\Vert }_{L_{x,y}^2}+\Vert W_{in}-(-{\partial }_y{b}_{in}){\Vert }_{H_x^{log}{L}_y^2}\le \epsilon {\nu }^{\frac{1}{2}}$, 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 $\Vert W_{\ne }{\Vert }_{L^2}\lesssim \epsilon {\nu }^{\frac{1}{2}}{e}^{-c{\nu }^{\frac{1}{3}}t}$ and $\Vert V_{\ne }{\Vert }_{L_t^2{L}_{x,y}^2}\lesssim \epsilon {\nu }^{\frac{1}{2}}$. 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.

53 Pages