On Liouville type theorems in the stationary non-Newtonian fluids.

In this paper we prove a Liouville type theorem for the stationary equations of a non-Newtonian fluid in R 3 with the viscous part of the stress tensor A p (u) = div (| D (u) | p − 2 D (u)) , where D (u) = 1 2 (∇ u + (∇ u) ⊤) and 9 5 < p < 3. We consider a weak solution u ∈ W l o c 1 , p (R 3) and its potential function V = (V i j) ∈ W l o c 2 , p (R 3) , i.e. ∇ ⋅ V = u. We show that there exists a constant s 0 = s 0 (p) such that if the L s mean oscillation of V for s > s 0 satisfies a certain growth condition at infinity, then the velocity field vanishes. Our result includes the previous results [5,6] as particular cases. [ABSTRACT FROM AUTHOR]