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.

中文翻译：

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无限处速度增长的轴对称Navier-Stokes方程的Liouville定理

在纸上科赫等。（2009年），作者做出以下推测：3D轴对称Navier–Stokes方程的任何有界古代温和解都是恒定的。并证明了在解无旋涡的情况下。本文的目的是通过允许解决方案以相对于原点的距离小于1的幂增长来提高结果。而且，我们将证明这种能力是证明Liouville型定理的最佳选择，因为我们可以找到Navier–Stokes方程的反例，使得如果解可以线性增长，则Liouville定理将失败。