Stability of Nontrivial Solutions of the Navier–Stokes System on the Three Dimensional Torus

Abstract

This paper examines the stability of nontrivial regular solutions to the Navier–Stokes equations on a three dimensional torus. It is shown that the W2, 1r-norm of the perturbation can be controlled if its initial data are small enough in the L2-norm. A key element of the proof is to apply the Marcinkiewicz theorem to obtain the estimate for the Stokes system. In particular we prove the stability of unforced two dimensional flows.