It has been an old and challenging problem to classify bounded ancient solutions of the incompressible Navier-Stokes equations, which could play a crucial role in the study of global regularity theory.
In the works (see the references), the authors made the following conjecture: { \it for the 3D axially symmetric Navier Stokes equations, bounded mild ancient solutions are constants}. In this article, we solve this conjecture in the case that $u$ is periodic in $z$. To the best of our knowledge, this seems to be the first result on this conjecture without unverified decay conditions. It also shows that nontrivial periodic solutions are not models of possible singularities or high velocity regions. Some partial results in the non-periodic case is also given.
