AI 中文总结
研究三维不可压缩纳维 - 斯托克斯方程的旋转向后自相似解，通过引入加权\(L^2\)框架，证明在特定条件下满足I型上界的此类解若旋转参数\(\alpha\)不当则为平凡解，部分回答佩雷尔曼问题，对旋转离散自相似解也有类似结果。
AI 中文摘要
我们考虑三维不可压缩纳维 - 斯托克斯方程的向后全局自相似解，其在缩放（自然抛物缩放）和绕给定轴以自相似时间的恒定角速度\(\alpha\)旋转的联合作用下是不变的。对于这些所谓的旋转自相似解（RSS），我们证明如果它们满足I型上界，并且旋转参数\(\alpha\)要么太小要么太大，那么它们必定是平凡的。这个刘维尔型结果将Nečas - Růžička - Šverák（'96）和Tsai（'98）仅考虑\(\alpha = 0\)的经典工作扩展到了经历非平凡旋转的相似剖面情况。我们的结果部分回答了佩雷尔曼提出的一个问题。对于在缩放和旋转的离散作用下不变的向后全局自相似解，即所谓的旋转离散自相似解（RDSS），在I型上界下，假设旋转参数\(\alpha\)取极值且缩放因子\(\lambda\)足够接近\(1\)时，我们得到了类似的刘维尔型结果。所有这些结果的证明基于引入一个强大 的加权\(L^2\)框架。特别是，我们的方法是定量的，并且对伯努利头压力是否满足最大值原理不敏感，这是先前工作中的一个关键障碍。
英文摘要
We consider backwards globally self-similar solutions of the 3D incompressible Navier-Stokes equations which are invariant under the joint action of scaling (the natural parabolic scaling) and rotation about a given axis, at a constant angular speed $α$ in self-similar time. For these so-called rotated self-similar solutions (RSS), we prove that if they satisfy a Type~I upper bound, and if the rotation parameter $α$ is either too small, or too large, then they must be trivial. This Liouville-type result extends the classical works of Nečas-Růžička-Šverák ('96) and Tsai ('98), which only consider $α=0$, to the case of similarity profiles which experience nontrivial rotation. Our results partially answer a question posed by Perelman. For backwards globally self-similar solutions which are invariant under the discrete action of scaling and rotation, the so-called rotated discretely self-similar solutions (RDSS), we obtain similar Liouville-type results under a Type~I upper bound, assuming extreme values of the rotation parameter $α$, and if the scaling factor $λ$ is sufficiently close to $1$. We also establish a new regularity criterion for 3D Navier-Stokes which is local in nature: if the solution satisfies a Type~I upper bound in a unit parabolic cylinder, and there is a single time-slice at which the solution is locally approximately self-similar, then the top-center of the parabolic cylinder is a regular point of the Navier-Stokes flow. The proof of all these results rests on the introduction of a robust weighted-$L^2$ framework. In particular, our method is quantitative and is not sensitive to whether the Bernoulli head pressure satisfies a maximum principle, which was a key obstruction in previous works.
Comments37 pages. Additional results and comments added
