In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R3\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb R^3$$\end{document}, having the diffusion term Ap(u)=∇·(|D(u)|p-2D(u))\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{A}}_p( u)=\nabla \cdot ( |{\varvec{D}}(u)|^{p-2} {\varvec{D}}(u))$$\end{document} with D(u)=12(∇u+(∇u)⊤)\documentclass[12pt]{minimal}
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\begin{document}$$ {\varvec{D}}(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top })$$\end{document}, 3/2<p<3\documentclass[12pt]{minimal}
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\begin{document}$$3/2<p< 3$$\end{document}. In the case 3/2<p≤9/5\documentclass[12pt]{minimal}
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\begin{document}$$3/2< p\le 9/5$$\end{document}, we show that a suitable weak solution u∈W1,p(R3)\documentclass[12pt]{minimal}
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\begin{document}$$u\in W^{1, p}(\mathbb R^3)$$\end{document} satisfying lim infR→∞|uB(R)|=0\documentclass[12pt]{minimal}
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\begin{document}$$ \liminf _{R \rightarrow \infty } |u_{ B(R)}| =0$$\end{document} is trivial, i.e., u≡0\documentclass[12pt]{minimal}
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\begin{document}$$u\equiv 0$$\end{document}. On the other hand, for 9/5<p<3\documentclass[12pt]{minimal}
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\begin{document}$$9/5<p<3$$\end{document} we prove the following Liouville type theorem: if there exists a matrix valued function V={Vij}\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{V}}= \{V_{ ij}\}$$\end{document} such that ∂jVij=ui\documentclass[12pt]{minimal}
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\begin{document}$$ \partial _jV_{ ij} =u_i$$\end{document}(summation convention), whose L3p2p-3\documentclass[12pt]{minimal}
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\begin{document}$$L^{\frac{3p}{2p-3}} $$\end{document} mean oscillation has the following growth condition at infinity, ∫-B(r)|V-VB(r)|3p2p-3dx≤Cr9-4p2p-3∀1<r<+∞,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\int \!\!\!\!\!\!-}_{B(r)} |{\varvec{V}}- {\varvec{V}}_{ B(r)} |^{\frac{3p}{2p-3}} \mathrm{d}x \le C r^{\frac{9-4p}{2p-3}}\quad \forall 1< r< +\infty , \end{aligned}$$\end{document}then u≡0\documentclass[12pt]{minimal}
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\begin{document}$$u\equiv 0$$\end{document}.
