{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:FX2KQXDMMJYTJ44GUCBP2T5KMY","short_pith_number":"pith:FX2KQXDM","schema_version":"1.0","canonical_sha256":"2df4a85c6c627134f386a082fd4faa663dd459fbdb77f3bc54bea7119bbbd88b","source":{"kind":"arxiv","id":"2607.18841","version":1},"attestation_state":"computed","paper":{"title":"A lecture on Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.HO","authors_text":"FAMSI), Sylvie Monniaux (I2M","submitted_at":"2026-07-21T08:26:46Z","abstract_excerpt":"The content of the following pages was part of a special topics lecture given at the Australian National University during the first semester of 2026. That was a very nice expericence, I really enjoyed giving this 12 weeks (2 hours a week) lecture. It is meant to be self contained, starting with results on Fourier transform and Sobolev spaces. As a toy model, before treating the Navier-Stokes system, we focus on the non linear heat equation where the non linearity is polynomial. Ultimately, we prove existence and uniqueness of mild solutions of the Navier-Stokes equations in critical spaces. T"},"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":"2607.18841","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.HO","submitted_at":"2026-07-21T08:26:46Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"9050369e1d8021a52b5bb268b04740a60be06ccb09f1db2b1e02eae17f0bcf0e","abstract_canon_sha256":"c50489cc21f77d41fd8a23b41890a9d7033f4cfe56fd380e4c80656c849b2e1a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-22T01:23:15.432972Z","signature_b64":"B4z2xl3R7W1lfl9kI1RLz2H8kV7CcrxyJjzrBMzcCB0X/OrzvZ+OfPS5kP5SA66C6+Kf8iT2T5/4q3GcFJPHDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2df4a85c6c627134f386a082fd4faa663dd459fbdb77f3bc54bea7119bbbd88b","last_reissued_at":"2026-07-22T01:23:15.432140Z","signature_status":"signed_v1","first_computed_at":"2026-07-22T01:23:15.432140Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A lecture on Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.HO","authors_text":"FAMSI), Sylvie Monniaux (I2M","submitted_at":"2026-07-21T08:26:46Z","abstract_excerpt":"The content of the following pages was part of a special topics lecture given at the Australian National University during the first semester of 2026. That was a very nice expericence, I really enjoyed giving this 12 weeks (2 hours a week) lecture. It is meant to be self contained, starting with results on Fourier transform and Sobolev spaces. As a toy model, before treating the Navier-Stokes system, we focus on the non linear heat equation where the non linearity is polynomial. Ultimately, we prove existence and uniqueness of mild solutions of the Navier-Stokes equations in critical spaces. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.18841","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.18841/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2607.18841","created_at":"2026-07-22T01:23:15.432575+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.18841v1","created_at":"2026-07-22T01:23:15.432575+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.18841","created_at":"2026-07-22T01:23:15.432575+00:00"},{"alias_kind":"pith_short_12","alias_value":"FX2KQXDMMJYT","created_at":"2026-07-22T01:23:15.432575+00:00"},{"alias_kind":"pith_short_16","alias_value":"FX2KQXDMMJYTJ44G","created_at":"2026-07-22T01:23:15.432575+00:00"},{"alias_kind":"pith_short_8","alias_value":"FX2KQXDM","created_at":"2026-07-22T01:23:15.432575+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/FX2KQXDMMJYTJ44GUCBP2T5KMY","json":"https://pith.science/pith/FX2KQXDMMJYTJ44GUCBP2T5KMY.json","graph_json":"https://pith.science/api/pith-number/FX2KQXDMMJYTJ44GUCBP2T5KMY/graph.json","events_json":"https://pith.science/api/pith-number/FX2KQXDMMJYTJ44GUCBP2T5KMY/events.json","paper":"https://pith.science/paper/FX2KQXDM"},"agent_actions":{"view_html":"https://pith.science/pith/FX2KQXDMMJYTJ44GUCBP2T5KMY","download_json":"https://pith.science/pith/FX2KQXDMMJYTJ44GUCBP2T5KMY.json","view_paper":"https://pith.science/paper/FX2KQXDM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.18841&json=true","fetch_graph":"https://pith.science/api/pith-number/FX2KQXDMMJYTJ44GUCBP2T5KMY/graph.json","fetch_events":"https://pith.science/api/pith-number/FX2KQXDMMJYTJ44GUCBP2T5KMY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FX2KQXDMMJYTJ44GUCBP2T5KMY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FX2KQXDMMJYTJ44GUCBP2T5KMY/action/storage_attestation","attest_author":"https://pith.science/pith/FX2KQXDMMJYTJ44GUCBP2T5KMY/action/author_attestation","sign_citation":"https://pith.science/pith/FX2KQXDMMJYTJ44GUCBP2T5KMY/action/citation_signature","submit_replication":"https://pith.science/pith/FX2KQXDMMJYTJ44GUCBP2T5KMY/action/replication_record"}},"created_at":"2026-07-22T01:23:15.432575+00:00","updated_at":"2026-07-22T01:23:15.432575+00:00"}