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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","authors_text":"Dongho Chae, Joerg Wolf","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-10T10:50:53Z","title":"On Liouville type theorems for the stationary MHD and Hall-MHD systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04495","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:82e43dbb4c712eacda72b37ad85ae4459e3568308c2921eac753f86cf29deea9","target":"record","created_at":"2026-05-17T23:58:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"856efb6c96f13d4280a9eff7001a354ca225318e80d87cce22e52c4971ab18d3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-12-10T10:50:53Z","title_canon_sha256":"e49e0dc52768e8a656e30be3cb9402495c6656ad39b6f34b858677c7c9c08a81"},"schema_version":"1.0","source":{"id":"1812.04495","kind":"arxiv","version":2}},"canonical_sha256":"c6ba99e9e9e80d0b0fc430e0311f377c5cf9d56e068c8bf4379b6fe54883c295","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c6ba99e9e9e80d0b0fc430e0311f377c5cf9d56e068c8bf4379b6fe54883c295","first_computed_at":"2026-05-17T23:58:01.301866Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:01.301866Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cNHpqTnHYSPy9qNGxbmDLJKAUS05Ef+U8CUdstkPa+iBw/VFPxUgdBf7tnaD0fGlpBqq1SddDo/l3D0XdW9gBA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:01.302246Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.04495","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:82e43dbb4c712eacda72b37ad85ae4459e3568308c2921eac753f86cf29deea9","sha256:14f96396c7fddc56d18b97c8bf28054be1edb304563be3f1f21144f9effda4b1"],"state_sha256":"7e2fc06466f969c0d68a528a413953fffe2e86af337bb362c0947f6077b667e3"}