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.

我们针对\（（-\ infty，0] \ times（{\ mathbb {R}} ^ 2 \ times {\ mathbb {T}上的轴对称不可压缩Navier–Stokes方程的有界轻度古代解建立Liouville定理} ^ 1）\），也就是在z方向上也是周期性的解，其结果受Chen等人的启发（对轴对称Navier–Stokes方程的爆破率具有较低的限制。） 。IMRN，8（art ID ID rnn016，31 pp），2008），Chen et al。（II Commun Partial Differ Equ 34（1-3）：203-232，2009）和Koch et al。（Acta Math 203（1） ）：83–105，2009），可以看作是完全解决\（（-infty，0] \ times {\ mathbb {R}} ^ 3 \）上的猜想的一步这是由Koch等人制作的。（Acta Math 203（1）：83–105，2009）描述了柯西问题的潜在奇点结构。在解决方案上没有进行未经验证的衰减假设。