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^\epsilon(x) = (v_0^h(x_\epsilon), \epsilon^{-1}v_0^n(x_\epsilon))^T,\quad x_\epsilon = (x_h, \epsilon x_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 $\epsilon > 0$, 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 $\epsilon$.