摘要
本文研究不可压缩Navier-Stokes方程的古代解所具有的Liouville性质.在二维情形以及三维轴对称具平凡角向速度(v_θ=0)情形下,本文证明了光滑的温和古代解的"最优"Liouville定理,即当涡度满足一定条件且速度场v关于空间变量次线性增长时,v恒为常向量,并且在速度场线性增长条件下给出了非平凡古代解的反例.其中,在二维情形下,涡度w需要满足的条件为,对所有的t∈(-∞,0)一致成立lim_(|x|→+∞)|w(x,t)|=0;在三维轴对称具平凡角向速度情形下,涡度w需要满足的条件为,对所有的t∈(-∞,0)一致成立lim_(r→+∞)(|w(x,t)|)/r=0.在三维轴对称具非平凡角向速度(v_θ≠0)的情形下,本文证明了,若Γ=rv_θ∈L_t~∞L_x^p(R^3×(-∞,0)),其中1≤p<∞,则有界的温和古代解必为常向量.
In this paper,we consider the Liouville property for ancient solutions of the incompressible NavierStokes equations.In 2D and 3D axially-symmetric cases without swirl,we prove the sharp Liouville theorems for smooth ancient mild solutions：Velocity fields v＇s are constants if vorticity fields satisfy certain condition and v＇s are sublinear with respect to the spatial variable,and we also give the counterexamples when v＇s are linear with respect to the spatial variable.The condition which vorticity fields need to satisfy is lim（｜x｜→＋∞）｜w（x,t）｜=0 and lim（｜w｜r→＋∞）｜w｜/（x1^2＋x2^2）^1/2=0 uniformly for all t∈（-∞,0） in 2D and 3D axially-symmetric cases without swirl,respectively.In the case when solutions are axially symmetric with nontrivial swirl,we prove that if Γ=rvθ∈Lt^∞Lx^p（R^3×（-∞,0））,where 1≤p∞,then bounded ancient mild solutions are constants.
作者
雷震
张旗
赵娜
LEI Zhen ZHANG Qi ZHAO Na
出处
《中国科学：数学》
CSCD
北大核心
2017年第10期1183-1198,共16页
Scientia Sinica：Mathematica
基金
国家自然科学基金(批准号:11421061和11222107)资助项目
