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

Abstract

In this paper we prove a Liouville type theorem for generalized stationary Navier-Stokes systems in $\Bbb R^3$, which model non-Newtonian fluids, where the Laplacian term $\Delta u$ is replaced by the corresponding non linear operator $\bA_p( u)=\nabla \cdot ( |\bD(u)|^{p-2} \bD(u))$ with $ \bD(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 $u\in W^{1, p}(\Bbb R^3)$ 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 impose the condition for the Liouville type theorem in terms of a potential function: if there exists a matrix valued potential function $\bV$ such that $ \nabla \cdot \bV =u$, whose $L^{\frac{3p}{2p-3}} $ mean oscillation has the following growth condition at infinity, $$ \intmw_{B(r)} |\bV- \bV_{ B(r)} |^{\frac{3p}{2p-3}} dx \le C r^{\frac{9-4p}{2p-3}}\quad \forall 1< r< +\infty, $$ then $u\equiv 0$. In the case of the Navier-Stokes equations, $p=2$, this improves the previous results in the literature.