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 $u\equiv 0$.