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

2018/12/10 by Dongho Chae, Chae, Dongho, Joerg Wolf +1
Mathematics · #35Q30 #76D03 #76D05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1812.04495

openalex publication_date 2018/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics(MHD) system in \Bbb R³. Let (v, B, p) be a smooth solution to the stationary MHD equations in \Bbb R³. We show that if there exist smooth matrix valued potential functions \bf Φ, \bf Ψ such that ∇ ⋅ \bf Φ =v and ∇ ⋅ \bf Ψ= B, whose L⁶ mean oscillations have certain growth condition near infinity, namely - ∫_B(r) |\mathbfΦ - \mathbfΦ _B(r) |⁶ dx + - ∫_B(r) |\mathbfΨ- \mathbfΨ _B(r) |⁶ dx≤ C r ∀ 1lt; rlt; +∞, then v=B= 0 and p=constant. With additional assumption of r^-8∫_B(r)|B-B_B(r)|⁶dx→ 0 as r→+∞, similar result holds also for the Hall-MHD system.

Citations

• Liouville theorems for the Navier–Stokes equations and applications

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