Improved Liouville theorems for axially symmetric Navier-Stokes equations

Abstract

In this paper, we consider the Liouville property for ancient solutions of the incompressible Navier-Stokes equations. In 2D and the 3D axially symmetric case without swirl, we prove sharp Liouville theorems for smooth ancient mild solutions: velocity fields $v$ are constants if vorticity fields satisfy certain condition and $v$ are sublinear with respect to spatial variables, and we also give counterexamples when $v$ are linear with respect to spatial variables. The condition which vorticity fields need to satisfy is $\underset{|x|\to +\infty }{\lim }|w(x,t)|=0$ and $\underset{r\to +\infty }{\lim }\frac{|w|}{\sqrt{x_1^2+x_2^2}}=0$ uniformly for all $t\in (-\infty ,0)$ in 2D and 3D axially symmetric case without swirl, respectively. In the case when solutions are axially symmetric with nontrivial swirl, we prove that if $\Gamma =r{v}_{\theta }\in L_t^{\infty }{L}_x^p({\mathbb{R}}^3\times (-\infty ,0))$ where $1\le p<\infty$, then bounded ancient mild solutions are constants.

22 pages

Cited by in corpus (1)

• On ancient periodic solutions to Axially-Symmetric Navier-Stokes Equations