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In this paper we prove a Liouville type theorem for the stationary equationsof a non-Newtonian fluid in $\mathbb{R}^3$ with the viscous part of the stresstensor $\mathbf{A}_p(u) = \mathrm{div} ( | \mathbf{D}(u) |^{p-2} \mathbf{D}(u))$, where $\mathbf{D}(u) = \frac 12 ( \nabla u + ( \nabla u )^{\top})$ and$\frac 95 < p < 3$. We consider a weak solution $u \inW^{1,p}_{loc}(\mathbb{R}^3)$ and its potential function $\mathbf{V} = (V_{ij})\in W^{2,p}_{loc}(\mathbb{R}^3)$, i.e. $\nabla \cdot \mathbf{V} = u$. We showthat there exists a constant $s_0=s_0(p)$ such that if the $L^s$ meanoscillation of $\mathbf{V}$ for $s>s_0$ satisfies a certain growth condition atinfinity, then the velocity field vanishes. Our result includes the previousresults \cite{CW20, CW19} as particular cases.

On Liouville type theorems in the stationary non-Newtonian fluids