Abstract

Consider the Cauchy problem of the Navier-Stokes equations on $ \Real^n$ with large initial data which is slowly varying in vertical variable.

We prove that in 4D space, the Navier-Stokes equations 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 $\epsilon$.