On Liouville type theorems in the stationary non-Newtonian fluids

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 $$\mathbf{A}$p(u) = $\mathrm{d}\mathrm{i}\mathrm{v}$ ( | $\mathbf{D}$(u) |^{p-2} $\mathbf{D}$(u) )$,where$$\mathbf{D}$(u) = \frac 12 ( \nabla u + ( \nabla u )^{\top})$and$\frac 95 < p < 3$\mathrm{.}Weconsideraweaksolution$u \in W^{1,p}{loc}($\mathbb{R}$^3)$anditspotentialfunction$$\mathbf{V}$ = (V_{ij}) \in W^{2,p}_{loc}($\mathbb{R}$^3)$,i\mathrm{.}e\mathrm{.}$\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.