Liouville theorem of axially symmetric Navier-Stokes equations with growing velocity at infinity

In the paper Koch et al. (2009), the authors make the following conjecture: any bounded ancient mild solution of the 3D axially symmetric Navier-Stokes equations is constant. And it is proved in the case that the solution is swirl free. Our purpose of this paper is to improve their result by allowing that the solution can grow with a power smaller than 1 with respect to the distance to the origin. Also, we will show that such a power is optimal to prove the Liouville type theorem since we can find counterexamples for the Navier-Stokes equations such that the Liouville theorem fails if the solution can grow linearly. (C) 2020 Elsevier Ltd. All rights reserved.