轴对称Navier-Stokes方程的Liouville型定理  

Liouville type theorems for axially symmetric Navier-Stokes equations

作　　者：雷震 任潇 张旗 Zhen Lei;Xiao Ren;Qi S.Zhang

机构地区：[1]复旦大学数学科学学院,上海200433 [2]Department of Mathematics,University of California,Riverside,Riverside,CA 92521,USA

出　　处：《中国科学：数学》2021年第6期971-984,共14页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11725102)资助项目。

摘　　要：不可压Navier-Stokes方程的有界古代解分类是一个古老而困难的问题,与Navier-Stokes方程整体正则性理论关系密切.特别地,有关于轴对称Navier-Stokes方程的如下Liouville型猜想:对于3维不可压轴对称Navier-Stokes方程,其有界古代解是常数.本文给出一种新的加权能量估计的方法,并在适当的Γ=rvθ收敛速率条件下得到Liouville定理;并且,用类似的能量估计,结合紧性方法,给出z-周期稳态解的Liouville定理的一个证明.本文的定理中不需要对速度场假设不自然的衰减速率条件.It has been an old and challenging problem to classify bounded ancient solutions of the incompressible Navier-Stokes equations,which could play a crucial role in the study of the global regularity theory.In particular,there is an important Liouville type conjecture in the axially symmetric case:for the 3 D axially symmetric NavierStokes equations,bounded mild ancient solutions are constants.In this paper,we propose a novel weighted energy method,and solve this problem under suitable convergence conditions for the swirl functionΓ=rvθ.Moreover,using similar estimates combined with compactness arguments,we prove that bounded mild ancient solutions which are periodic in z must be constants.We do not assume any unverified decay conditions on the velocity field.

关 键 词：NAVIER-STOKES方程 古代解 轴对称 LIOUVILLE定理 

分 类 号：O175[理学—数学]