{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:ZKDW5M2UPZNXVTRDEPNKKZRMFJ","short_pith_number":"pith:ZKDW5M2U","schema_version":"1.0","canonical_sha256":"ca876eb3547e5b7ace2323daa5662c2a73dbe6546818c3e30e47ebf02d49b791","source":{"kind":"arxiv","id":"1605.08135","version":2},"attestation_state":"computed","paper":{"title":"Limitations of the background field method applied to Rayleigh-B\\'enard convection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Camilla Nobili, Felix Otto","submitted_at":"2016-05-26T03:21:18Z","abstract_excerpt":"We consider Rayleigh-B\\'enard convection as modeled by the Boussinesq equations, in case of infinite Prandtl number. There is a broad interest in bounds of the upwards heat flux, as given by the Nusselt number ${\\rm Nu}$, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime ${\\rm Ra}\\gg 1$. In several works, the background field method applied to the temperature field has been used to provide upper bounds on ${\\rm Nu}$ in terms of ${\\rm Ra}$. In these applications, the background field method comes in form of a variational prob"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1605.08135","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-26T03:21:18Z","cross_cats_sorted":[],"title_canon_sha256":"52fcb4bd9de33b4a5bb48cd0fb17d9d668cf3ccf88844ca2c9172a8b0dce3d25","abstract_canon_sha256":"ff905018c7d257e98b0199c6ede9f672f13b3244f7362aba23c21579e48f07cd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:23.659858Z","signature_b64":"+M0EcBkGBJSbvzeMW6ogdYTRB5DmL9ZYj3lnhQnwETknAB3EnrkEFbkS9bXHvjs11sBY68psqxc2BE7g+O/6Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca876eb3547e5b7ace2323daa5662c2a73dbe6546818c3e30e47ebf02d49b791","last_reissued_at":"2026-05-18T00:33:23.659236Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:23.659236Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Limitations of the background field method applied to Rayleigh-B\\'enard convection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Camilla Nobili, Felix Otto","submitted_at":"2016-05-26T03:21:18Z","abstract_excerpt":"We consider Rayleigh-B\\'enard convection as modeled by the Boussinesq equations, in case of infinite Prandtl number. There is a broad interest in bounds of the upwards heat flux, as given by the Nusselt number ${\\rm Nu}$, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime ${\\rm Ra}\\gg 1$. In several works, the background field method applied to the temperature field has been used to provide upper bounds on ${\\rm Nu}$ in terms of ${\\rm Ra}$. In these applications, the background field method comes in form of a variational prob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.08135","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1605.08135","created_at":"2026-05-18T00:33:23.659329+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.08135v2","created_at":"2026-05-18T00:33:23.659329+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.08135","created_at":"2026-05-18T00:33:23.659329+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZKDW5M2UPZNX","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZKDW5M2UPZNXVTRD","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZKDW5M2U","created_at":"2026-05-18T12:30:53.716459+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZKDW5M2UPZNXVTRDEPNKKZRMFJ","json":"https://pith.science/pith/ZKDW5M2UPZNXVTRDEPNKKZRMFJ.json","graph_json":"https://pith.science/api/pith-number/ZKDW5M2UPZNXVTRDEPNKKZRMFJ/graph.json","events_json":"https://pith.science/api/pith-number/ZKDW5M2UPZNXVTRDEPNKKZRMFJ/events.json","paper":"https://pith.science/paper/ZKDW5M2U"},"agent_actions":{"view_html":"https://pith.science/pith/ZKDW5M2UPZNXVTRDEPNKKZRMFJ","download_json":"https://pith.science/pith/ZKDW5M2UPZNXVTRDEPNKKZRMFJ.json","view_paper":"https://pith.science/paper/ZKDW5M2U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.08135&json=true","fetch_graph":"https://pith.science/api/pith-number/ZKDW5M2UPZNXVTRDEPNKKZRMFJ/graph.json","fetch_events":"https://pith.science/api/pith-number/ZKDW5M2UPZNXVTRDEPNKKZRMFJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZKDW5M2UPZNXVTRDEPNKKZRMFJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZKDW5M2UPZNXVTRDEPNKKZRMFJ/action/storage_attestation","attest_author":"https://pith.science/pith/ZKDW5M2UPZNXVTRDEPNKKZRMFJ/action/author_attestation","sign_citation":"https://pith.science/pith/ZKDW5M2UPZNXVTRDEPNKKZRMFJ/action/citation_signature","submit_replication":"https://pith.science/pith/ZKDW5M2UPZNXVTRDEPNKKZRMFJ/action/replication_record"}},"created_at":"2026-05-18T00:33:23.659329+00:00","updated_at":"2026-05-18T00:33:23.659329+00:00"}