Stability of 2D incompressible flows in ${\mathbf{R}}^3$

Abstract

We investigate the global in time stability of regular solutions with large velocity vectors to the evolutionary Navier-Stokes equation in ${\mathbf{R}}^3$. The class of stable flows contains all two dimensional weak solutions. The only assumption which is required is smallness of the $L_2$-norm of initial perturbation or its derivative with respect to the `$z$'-coordinate in the same norm. The magnitude of the rest of the norm of initial datum is not restricted.