On Liouville type theorems in the stationary non-Newtonian fluids

TL;DR

This paper establishes a Liouville type theorem for stationary non-Newtonian fluid equations in three dimensions, showing under certain growth conditions on the potential function that the velocity must vanish, extending previous results.

Contribution

It proves a new Liouville theorem for non-Newtonian fluids with specific stress tensor conditions, including broader growth conditions and potential function analysis.

Findings

Velocity field vanishes under growth conditions

Includes previous Liouville results as special cases

Extends understanding of stationary non-Newtonian fluids

Abstract

In this paper we prove a Liouville type theorem for the stationary equations of a non-Newtonian fluid in $\mathbb{R}^3$ with the viscous part of the stress tensor $\mathbf{A}_p(u) = \mathrm{div} ( | \mathbf{D}(u) |^{p-2} \mathbf{D}(u) )$, where $\mathbf{D}(u) = \frac 12 ( \nabla u + ( \nabla u )^{\top})$ and $\frac 95 < p < 3$. We consider a weak solution $u \in W^{1,p}_{loc}(\mathbb{R}^3)$ and its potential function $\mathbf{V} = (V_{ij}) \in W^{2,p}_{loc}(\mathbb{R}^3)$, i.e. $\nabla \cdot \mathbf{V} = u$. We show that there exists a constant $s_0=s_0(p)$ such that if the $L^s$ mean oscillation of $\mathbf{V}$ for $s>s_0$ satisfies a certain growth condition at infinity, then the velocity field vanishes. Our result includes the previous results \cite{CW20, CW19} as particular cases.