轴对称Navier-Stokes方程改进的Liouville定理

发布时间：2018-06-18 01:04

  本文选题：Navier-Stokes方程 + 古代解　； 参考：《中国科学:数学》2017年10期

【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.
[Abstract]:In this paper, we study the Liouville properties of the ancient solutions of incompressible Navier-Stokes equations. In the case of two-dimensional and three-dimensional axisymmetric angular velocities v _ 胃 _ 0, we prove the "optimal" Liouville theorem for smooth and mild ancient solutions. That is, when the vorticity satisfies certain conditions and the velocity field v increases sublinearly with respect to spatial variables, v is constant vector, and a counterexample of the nontrivial ancient solution is given under the condition of linear growth of velocity field. Where, in two-dimensional case, the condition that vorticity w needs to be satisfied is that for all t 鈭，

本文编号：2033284