A Liouville theorem for Axi-symmetric Navier–Stokes equations on $${\mathbb {R}}^2 \times {\mathbb {T}}^1$$ R 2 × T 1

We establish a Liouville theorem for bounded mild ancient solutions to the axi-symmetric incompressible Navier–Stokes equations on $$(-\infty , 0] \times ({\mathbb {R}}^2 \times {\mathbb {T}}^1)$$ , i.e. those solutions which are also periodic in z direction. The result, inspired by the works Chen et al. (Lower bound on the blow-up rate of the axisymmetric Navier–Stokes equations. International Mathematics Research Notices. IMRN, 8(artical ID rnn016, 31 pp), 2008), Chen et al. (II Commun Partial Differ Equ 34(1–3):203–232, 2009) and Koch et al. (Acta Math 203(1):83–105, 2009), can be regarded as a step forward to completely solve the conjecture on $$(-\infty , 0] \times {\mathbb {R}}^3$$ which was made in Koch et al. (Acta Math 203(1):83–105, 2009) to describe the potential singularity structures of the Cauchy problem. No unverified decay assumption is made on the solutions.