On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations.

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R 3 , having the diffusion term A p (u) = ∇ · (| D (u) | p - 2 D (u)) with D (u) = 1 2 (∇ u + (∇ u) ⊤) , 3 / 2 < p < 3 . In the case 3 / 2 < p ≤ 9 / 5 , we show that a suitable weak solution u ∈ W 1 , p (R 3) satisfying lim inf R → ∞ | u B (R) | = 0 is trivial, i.e., u ≡ 0 . On the other hand, for 9 / 5 < p < 3 we prove the following Liouville type theorem: if there exists a matrix valued function V = { V ij } such that ∂ j V ij = u i (summation convention), whose L 3 p 2 p - 3 mean oscillation has the following growth condition at infinity, ∫ - B (r) | V - V B (r) | 3 p 2 p - 3 d x ≤ C r 9 - 4 p 2 p - 3 ∀ 1 < r < + ∞ , then u ≡ 0 . [ABSTRACT FROM AUTHOR]