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

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 Phi, Psi such that del. Phi = v and del. Psi = B, whose L-6 mean oscillations have certain growth condition near infinity, namely -integral(B(r)) vertical bar Phi - Phi(B(r))vertical bar(6)dx + -integral(B(r)) vertical bar Psi - Psi(B(r))vertical bar(6)dx <= Cr for all 1 < r < +infinity, then v = B = 0 and p = constant. With additional assumption of r(-8) integral(B(r)) vertical bar B - B-B(r)vertical bar(6)dx -> 0 as r -> +infinity, similar result holds also for the Hall-MHD system. (C) 2021 Elsevier Inc. All rights reserved.