slab 中稳态 Navier-Stokes 系统解的刘维尔型定理
偏微分方程分析
2022-08-22 v4
摘要
我们研究了三维 slab 中具有无滑移边界条件或周期边界条件的稳态不可压缩 Navier-Stokes 系统解的刘维尔型定理。当规定无滑移边界条件时，我们证明任何有界解若是轴对称的或 有界则为平凡解，并且当速度在 空间中不大时，一般三维解必为泊肃叶流。当 slab 边界上施加周期边界条件时，我们证明有界解若其旋流或径向速度独立于角变量，或 随 趋于无穷而衰减至零，则必为常向量。证明基于方程的基本结构与能量估计。关键技术是建立刻画非平凡解的 Dirichlet 积分增长的 Saint-Venant 型估计。
引用
@article{arxiv.2205.13259,
title = {Liouville-type theorems for steady solutions to the Navier-Stokes system in a slab},
author = {Jeaheang Bang and Changfeng Gui and Yun Wang and Chunjing Xie},
journal= {arXiv preprint arXiv:2205.13259},
year = {2022}
}
备注
Theorem 1.4 has been updated where Liouville type theorem for flows in periodic slab was proved as long as L^\infty norm of the velocity field is not so big. The bound for the velocity in Theorems 1.3 is presented in a more precise way