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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}, when the initial data is close to stable monotone shear flows. We prove the asymptotic stability and obtain the sharp stability threshold nu 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu <^>{\frac{1}{2}}$$\end{document} for perturbations in the critical space HxlogLy2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{log}_xL<^>2_y$$\end{document}. Specifically, if the initial velocity Vin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{in}$$\end{document} and the corresponding vorticity Win\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{in}$$\end{document} are nu 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu <^>{\frac{1}{2}}$$\end{document}-close to the shear flow (bin(y),0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(b_{in}(y),0)$$\end{document} in the critical space, i.e., & Vert;Vin-(bin(y),0)& Vert;Lx,y2+& Vert;Win-(-partial derivative ybin)& Vert;HxlogLy2 <=epsilon nu 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert V_{in}-(b_{in}(y),0)\Vert _{L_{x,y}<^>2}+\Vert W_{in}-(-\partial _yb_{in})\Vert _{H<^>{log}_xL<^>2_y}\le \varepsilon \nu <^>{\frac{1}{2}}$$\end{document}, then the velocity V(t) stay nu 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu <^>{\frac{1}{2}}$$\end{document}-close to a shear flow (b(t, y), 0) that solves the free heat equation (partial derivative t-nu partial derivative yy)b(t,y)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\partial _t-\nu \partial _{yy})b(t,y)=0$$\end{document}. We also prove the enhanced dissipation and inviscid damping, namely, the nonzero modes of vorticity and velocity decay in the following sense & Vert;W not equal & Vert;L2 less than or similar to epsilon nu 12e-c nu 13t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert W_{\ne }\Vert _{L<^>2}\lesssim \varepsilon \nu <^>{\frac{1}{2}}e<^>{-c\nu <^>{\frac{1}{3}}t}$$\end{document} and & Vert;V not equal & Vert;Lt2Lx,y2 less than or similar to epsilon nu 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert V_{\ne }\Vert _{L<^>2_tL<^>2_{x,y}}\lesssim \varepsilon \nu <^>{\frac{1}{2}}$$\end{document}. In the proof, we construct a time-dependent wave operator corresponding to the Rayleigh operator b(t,y)Id-partial derivative yyb(t,y)Delta-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b(t,y)\textrm{Id}-\partial _{yy}b(t,y)\Delta <^>{-1}$$\end{document}, which could be useful in future studies.