On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations

Abstract

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 $$u\equiv 0$$.