In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in \(\mathbb R^3\), having the diffusion term \({\varvec{A}}_p( u)=\nabla \cdot ( |{\varvec{D}}(u)|^{p-2} {\varvec{D}}(u))\) with \( {\varvec{D}}(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top })\), \(3/2<p< 3\). In the case \(3/2< p\le 9/5\), we show that a suitable weak solution \(u\in W^{1, p}(\mathbb R^3)\) satisfying \( \liminf _{R \rightarrow \infty } |u_{ B(R)}| =0\) is trivial, i.e., \(u\equiv 0\). On the other hand, for \(9/5<p<3\) we prove the following Liouville type theorem: if there exists a matrix valued function \({\varvec{V}}= \{V_{ ij}\}\) such that \( \partial _jV_{ ij} =u_i\)(summation convention), whose \(L^{\frac{3p}{2p-3}} \) mean oscillation has the following growth condition at infinity,$$\begin{aligned} {\int \!\!\!\!\!\!-}_{B(r)} |{\varvec{V}}- {\varvec{V}}_{ B(r)} |^{\frac{3p}{2p-3}} \mathrm{d}x \le C r^{\frac{9-4p}{2p-3}}\quad \forall 1< r< +\infty , \end{aligned}$$then \(u\equiv 0\).

中文翻译：

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平稳非牛顿流体方程的Liouville型定理

在本文中，我们证明了\（\ mathbb R ^ 3 \）中具有扩散项\（{\ varvec {A}} _ p（u）= \ nabla \ cdot（ | {\ varvec {D}}（u）| ^ {p-2} {\ varvec {D}}（u））\）与\（{\ varvec {D}}（u）= \ frac {1} {2}（\ nabla u +（\ nabla u）^ {\ top}）\），\（3/2 <p <3 \）。在\（3/2 <p \ le 9/5 \）的情况下，我们证明了满足\ {\的合适的弱解\（u \ in W ^ {1，p}（\ mathbb R ^ 3）\）liminf _ {R \ rightarrow \ infty} | u_ {B（R）} | = 0 \）是微不足道的，即\（u \ equiv 0 \）。另一方面，对于\（9/5 <p <3 \）我们证明下面的Liouville型定理：如果存在矩阵值函数\（{\ varvec {V}} = \ {V_ {ij} \} \），使得\（\ partial _jV_ {ij} = u_i \）（总结惯例），其\（L ^ {\ frac {3p} {2p-3}} \）的平均振荡在无穷大处具有以下增长条件$$ \ begin {aligned} {\ int \！\！\！\ ！\！\！-} _ {B（r）} | {\ varvec {V}}-{\ varvec {V}} _ {B（r）} | ^ {\ frac {3p} {2p-3} } \ mathrm {d} x \ le C r ^ {\ frac {9-4p} {2p-3}} \ quad \ forall 1 <r <+ \ infty，\ end {aligned} $$然后\（u \等于0 \）。