On Liouville type theorems for the stationary MHD and Hall-MHD systems

Abstract

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics (MHD) system in R 3 . Let ( v , B , p ) be a smooth solution to the stationary MHD equations in R 3 . We show that if there exist smooth matrix valued potential functions Φ , Ψ such that ∇ ⋅ Φ = v and ∇ ⋅ Ψ = B , whose L 6 mean oscillations have certain growth condition near infinity, namely ⨍ B ( r ) | Φ − Φ B ( r ) | 6 d x + ⨍ B ( r ) | Ψ − Ψ B ( r ) | 6 d x ≤ C r ∀ 1 < r < + ∞ , then v = B = 0 and p =constant. With additional assumption of r − 8 ∫ B ( r ) | B − B B ( r ) | 6 d x → 0 as r → + ∞ , similar result holds also for the Hall-MHD system.