Global Regularity to the Navier-Stokes Equations for A Class of Large Initial Data

Abstract

We prove that for initial data of the form \begin{equation}\nonumber u_0^ε(x) = (v_0^h(x_ε), ε^{-1}v_0^n(x_ε))^T,\quad x_ε= (x_h, εx_n)^T, n \geq 4, \end{equation} the Cauchy problem of the incompressible Navier-Stokes equations on ${\mathbb{R}}^n$ is globally well-posed for all small $\hat{I}\mu >0$, provided that the initial velocity profile $v_0$ is analytic in $x_n$ and certain norm of $v_0$ is sufficiently small but independent of $\hat{I}\mu$.