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A Liouville theorem for Axi-symmetric Navier–Stokes equations on $${\mathbb {R}}^2 \times {\mathbb {T}}^1$$ R 2 × T 1
Mathematische Annalen ( IF 1.3 ) Pub Date : 2021-01-17 , DOI: 10.1007/s00208-020-02128-9
Zhen Lei , Xiao Ren , Qi S. Zhang

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.



中文翻译:

$ {{\ mathbb {R}} ^ 2 \ times {\ mathbb {T}} ^ 1 $$ R 2×T 1上的轴对称Navier–Stokes方程的Liouville定理

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。

更新日期:2021-01-18
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