On Liouville type theorems in the stationary non-Newtonian fluids

Abstract

In this paper we prove a Liouville type theorem for the stationary equations of a non-Newtonian fluid in ${\mathbb{R}}^3$ with the viscous part of the stress tensor ${\mathbf{A}}_p(u)=\mathrm{div}(|\mathbf{D}(u){|}^{p-2}\mathbf{D}(u))$, where $\mathbf{D}(u)=\frac{1}{2}(\nabla u+(\nabla u{)}^{\top })$ and $\frac{9}{5}<p<3$. We consider a weak solution $u\in W_{loc}^{1,p}({\mathbb{R}}^3)$ and its potential function $\mathbf{V}=(V_{ij})\in W_{loc}^{2,p}({\mathbb{R}}^3)$, i.e. $\nabla \cdot \mathbf{V}=u$. We show that there exists a constant $s_0=s_0(p)$ such that if the $L^s$ mean oscillation of $\mathbf{V}$ for $s>s_0$ satisfies a certain growth condition at infinity, then the velocity field vanishes. Our result includes the previous results \cite{CW20, CW19} as particular cases.

17 pages