On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations

초록

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R3, having the diffusion term Ap(u) = ∇ · (| D(u) | p - 2D(u)) with D(u)=12(∇u+(∇u)⊤), 3 / 2 < p< 3. In the case 3 / 2 < p≤ 9 / 5 , we show that a suitable weak solution u∈ W1 , p(R3) satisfying lim inf R → ∞| uB ( 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= {Vij} such that ∂jVij= ui(summation convention), whose L3p2p-3 mean oscillation has the following growth condition at infinity, ∫-B(r)|V-VB(r)|3p2p-3dx≤Cr9-4p2p-3∀1<r<+∞,then u≡ 0. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.

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