Remarks on time dependent periodic Navier-Stokes flows on a two-dimensional torus

This note is a continuation of the earlier paper [Chen, Z. M., Price, W. G.: Time dependent periodic Navier–Stokes flows on a two-dimensional torus. Commun. Math. Phys. {\bf 179}, 577–597 (1996)], where the evidences of the occurrence for the time dependent periodic Navier–Stokes flows were found based on a combination of analysis and computation. This investigation is now confirmed via rigorous analysis. The existence of the time dependent periodic Navier–Stokes flows on a two-dimensional torus is prove