轴对称Navier-Stokes方程的Liouville型定理

Liouville type theorems for axially symmetric Navier-Stokes equations

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.