On Liouville type theorem for a generalized stationary Navier-Stokes equations

In this paper we prove a Liouville type theorem for generalized stationary Navier-Stokes systems in ${\mathbb{R}}^3$, which model non-Newtonian fluids, where the Laplacian term $\mathrm{\Delta }u$ is replaced by the corresponding non linear operator ${\text{bA}}_p(u)=\mathrm{\nabla }\cdot (\mathrm{\mid }\text{bD}(u){\mathrm{\mid }}^{p-2}\text{bD}(u))$ with $ \bD(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top})$,$3/2<p< 3$\mathrm{.}Inthecase$3/2< p\le 9/5$weshowthatasuitableweaksolution$u\in W^{1, p}(\Bbb R^3)$satisfying$ \liminf_{R \rightarrow \infty} |u_{ B®}| =0$istrivial,i\mathrm{.}e\mathrm{.}$u\equiv 0$\mathrm{.}Ontheotherhand,for$9/5<p<3$ we impose the condition for the Liouville type theorem in terms of a potential function: if there exists a matrix valued potential function $\text{bV}$ such that $\mathrm{\nabla }\cdot \text{bV}=u$, whose $L^{\frac{3p}{2p-3}}$ mean oscillation has the following growth condition at infinity, $${\text{intmw}}_{B(r)}\mathrm{\mid }\text{bV}-{\text{bV}}_{B(r)}{\mathrm{\mid }}^{\frac{3p}{2p-3}}dx\le C{r}^{\frac{9-4p}{2p-3}}\mathrm{\forall }1<r<+\mathrm{\infty },$$ then $u\equiv 0$. In the case of the Navier-Stokes equations, $p=2$, this improves the previous results in the literature.