{"paper":{"title":"On Liouville type theorems for the stationary MHD and Hall-MHD systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dongho Chae, Joerg Wolf","submitted_at":"2018-12-10T10:50:53Z","abstract_excerpt":"In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics(MHD) system in $\\Bbb R^3$. Let $(v, B, p)$ be a smooth solution to the stationary MHD equations in $\\Bbb R^3$. We show that if there exist smooth matrix valued potential functions ${\\bf \\Phi}$, ${\\bf \\Psi}$ such that $ \\nabla \\cdot {\\bf \\Phi} =v$ and $\\nabla \\cdot {\\bf \\Psi}= B$, whose $L^6$ mean oscillations have certain growth condition near infinity, namely $$-\\!\\!\\!\\!\\!\\int_{B(r)} |\\mathbf{\\Phi} - \\mathbf{\\Phi}_{ B(r)} |^6 dx + -\\!\\!\\!\\!\\!\\int_{B(r)} |\\mathbf{\\Psi}- \\mathbf{\\Psi}_{ B(r)} |^6 dx\\le C r\\q"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04495","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}