Remarks on Liouville type theorems for the 3D steady axially symmetric Navier–Stokes equations

Abstract

In this note, we investigate the 3D steady axially symmetric Navier-Stokes equations, and obtained Liouville type theorems if the velocity or the vorticity satisfies some a priori decay assumptions.

Liouville theorem for steady-state solutions of simplified Ericksen-Leslie system

“…For 2-D and axially-symmetric cases, one can refer [19,20,21] and the references therein for more details. Recently, Wendong Wang [22] and Na Zhao [23] obtained the Liouville Theorem with the conditions that v is axially symmetric and |v(x 1 , x 2 , x 3 )| ≤ C/(1 + r′) α , where α > 2 3 and r′ = x 2 1 + x 2 2 . When u ≡ 0, system (1.1) becomes the harmonic maps:…”

A Liouville type theorem for axially symmetric D-solutions to steady Navier–Stokes equations

“…Also we remark that we finished the result in April, 2018. Afterwards, we saw the paper by Wang Wendong [12] in arXiv, who proved a similar result independently. Throughout this paper, we will use…”

Liouville type theorems for stationary Navier–Stokes equations

“…0, then the above shows lim j!1 R jx 0 j\R j jruj 2 ¼ 0, and hence u 0. Alternatively, if condition (19) holds that there is a sequence R j ! 1 as j !…”

“…There is also a rich literature on the Liouville problem for the subclass of axisymmetric solutions. As we will not discuss it here, we only refer to [1,10,19] and their references.…”