{"as_of":"2026-08-11T20:28:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:689cf1ba02562bbbefc9ef76fedaef04c8ff02a8687b9552655da82fbe6eac74","coverage":[{"denominator":32,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":32,"source":"paper_references, paper_reference_links","source_observed_at":"2026-07-12T07:22:30.053977Z","state":"measured"},{"denominator":32,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":32,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-11T06:34:44.6726+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2607.02739/citation-record","integrity":"/paper/2607.02739/integrity","json":"/paper/2607.02739/citation-record.json","paper":"/paper/2607.02739"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Optimization Algorithms on Matrix Manifolds","venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:b19452585f4cb08e8a55fe48c57878b29535407db20d1393b375a7720393ad2b","observation_id":"be427df2-d182-4a36-b2ae-e8e2f02aa466","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Sobolev Spaces","venue":null,"work_id":null,"year":2005},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:07138ffd21cae52ed03d0eb5a44809bef82e97d599ef96a26dd88314e905be6b","observation_id":"5f7221e5-7561-4fdc-afeb-42e27555e2df","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Sharp bounds on enstrophy growth for viscous scalar conservation laws","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:38d1cdb2752611afbfa5f7182ba36fed72c752fb306ca1d19889a42efdcf4b91","observation_id":"e66d13ed-05b5-4ebb-8d87-264f751f4246","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Maximum palinstrophy amplification in the two-dimensional navier–stokes equations","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:6019559d692fdbb673f4820b6d3847713f215b4e5bc15f7782d2b75ed16935b0","observation_id":"e0a0cf48-44a4-4f4e-a2cb-0c1c5cf73225","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/j.physd.2011.07.003","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"On maximum enstrophy growth in a hy- drodynamic system","venue":"Physica D Nonlinear Phenomena","work_id":"283c329e-5931-4fa2-b7c8-15191c99fc65","year":2011},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:eb6e8de0849a32a3ca14c1588e970fc5b29eb0c6d71d803eee02f5f9b077870f","observation_id":"d4537372-d8b3-450e-abdd-b36a87b1c020","resolution":{"observed_at":"2026-07-12T07:28:35.152024Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-11T06:34:44.6726+00:00","source":"crossref"},{"observed_at":"2026-08-11T06:34:36.301508+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Maximum palinstrophy growth in 2d in- compressible flows","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:817dc89eb215e04d3f1a3e97657c7329a59602df0f73e0f90e0adb9368123254","observation_id":"d7ca8419-6120-4fdd-bfc4-1160687fda67","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1017/jfm.2017.136","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Extreme vortex states and the growth of enstrophy in three-dimensional incompressible flows","venue":"Journal of Fluid Mechanics","work_id":"159ca34a-7dff-459a-af27-d418beb7bfac","year":2017},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:fdd993c2b190ecdb436bf2c7b013683391a421b83d95ada95430c6171d6297fa","observation_id":"30954059-ea04-42cf-8f01-cb78b3129f7f","resolution":{"observed_at":"2026-07-12T07:28:35.175033Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-11T06:34:44.6726+00:00","source":"crossref"},{"observed_at":"2026-08-11T06:34:36.301508+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Nonlinearity and Functional Analysis","venue":null,"work_id":null,"year":1977},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:33e624941c83dc5f40e408b8fe7475e820dbd796749325d3daacfc12d23a8ecc","observation_id":"7493cbbd-ebc8-43ad-8297-261507e29812","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Spectral Methods","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:5f2147ab27a3405f1b7a67d1c35851be657dfe0368b512acd7aa86d76edc13dd","observation_id":"c4f82331-50cf-412f-9e5b-17893a12639c","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"L3,∞-solutions of the Navier-Stokes equations and backward uniqueness","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:ec2bb04c4634a2647ca1a77a907158e7252b233323aa5feae5497a43e31d3a63","observation_id":"ba9b0dac-c6f9-4fca-8ea0-9813808a4c28","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Existence and smoothness of the Navier–Stokes equation","venue":null,"work_id":null,"year":2000},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:9f78949721232291d55895bc119411019dd8a85475273fc4f80dd08c685ea415","observation_id":"25f71443-a0d2-4d62-a8f8-4958b980ad51","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/0022-1236(89)90015-3","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Gevrey class regularity for the solutions of the navier-stokes equations","venue":"Journal of Functional Analysis","work_id":"4d28e95b-f0e9-42ff-817c-229fb1e55d7c","year":1989},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:ee431206f14fa6ca7bbe19ff47d6653972679534a8d7b20b7901d022350f589b","observation_id":"a8de983d-2555-4ebc-90e5-b280eb90a9f5","resolution":{"observed_at":"2026-07-12T07:28:35.140130Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-11T06:34:44.6726+00:00","source":"crossref"},{"observed_at":"2026-08-11T06:34:36.301508+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Potentially singular behavior of the 3D Navier–Stokes equations","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:f9301e5f71c344119ee0834256f6cd72be451a30ed090e3f99d927701ffe25ac","observation_id":"f3850625-05a8-4eb9-8931-f48addb3bb74","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Searching for singularities in navier–stokes flows based on the ladyzhenskaya–prodi–serrin conditions","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:9e74ac27837326b201f9d52e5df57917f32bc541408e9ed32ff642bf8f7c0f0c","observation_id":"7dfefb3e-8020-413d-a5f4-ba49ebd747dc","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Maximum amplification of enstro- phy in three-dimensional Navier-Stokes flows","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:3e4bf8308d48996598e87c9a370f2cf82f674c9cc18d6917ef9612b14395362e","observation_id":"d0050bce-069a-4408-9f80-f1ef9e3f3647","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"On the existence and unique- ness of the solution of the nonstationary problem for a viscous, incom- pressible fluid","venue":null,"work_id":null,"year":1957},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:2cebee9a7d7e2e0cc7be17c6596065fb006bfb58a9e721361aa48dc8cb206857","observation_id":"28cf21bf-b738-4627-a6d7-e92a8b454b2d","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1512/iumj.2008.57.3716","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Limits on enstrophy growth for solutions of the three-dimensional Navier-Stokes equations","venue":"Indiana University Mathematics Journal","work_id":"e447de15-1f4c-49b8-96c6-3face22973e3","year":2008},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:25227f70b5125d019696a066a73a4c4b7b9f8f2b3bcb1e9961ed543ce9a8393b","observation_id":"fe5f3fa4-92f5-456c-a980-46203d307897","resolution":{"observed_at":"2026-07-12T07:28:35.150344Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-11T06:34:44.6726+00:00","source":"crossref"},{"observed_at":"2026-08-11T06:34:36.301508+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Vorticity and Incompressible Flow","venue":null,"work_id":null,"year":2002},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:18ed7557b123d2ec7bb2c7b6db5f6347d3bad0b5e8c110aea5aacc4d60230d24","observation_id":"e21fe9fb-012b-4a76-8d23-7339ac0f2e00","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Sobolev Gradients and Differential Equations","venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:daf0a20bd369768eea1bd23c96d566e533811fb1859d8b2e8a83afeece5439bc","observation_id":"f5527eb0-2da9-4c68-af0e-414666133caf","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Numerical Optimization","venue":null,"work_id":null,"year":1999},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:ff6a8e7333c3d40f309b30275b19dfb58e74ade9056a988a81be29406c15ddac","observation_id":"b8b0ee1b-0fd1-471e-8bb2-457beb728898","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"NumericalOptimization","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:5ae1e7052ddad7d5ea5a1c0b2227c4a41a5798314c7aaf48dc235a9c9a2d3f12","observation_id":"e703efb3-a1c7-415e-b017-8c7b34cd04f5","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Numerical Recipes","venue":null,"work_id":null,"year":2007},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:449f288117f0f7d57b1d270e119e9d3f796f0a55af334f8c2280e8a6f1ec1133","observation_id":"81f466a4-2252-4d18-9370-9c386398c245","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Un teorema di unicità per le equazioni di navier-stokes","venue":null,"work_id":null,"year":1959},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:c761aa17aebe7f683b0f4576196ec23e836bca698ea3f41d8c3999a8c817d41d","observation_id":"344e7c45-4f5b-49ef-a262-c255712938c8","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/j.jcp.2008.03.013","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Adjoint-based optimization of PDE systems with al- ternative gradients","venue":"Journal of Computational Physics","work_id":"f34a68c1-e391-4a5b-9d58-eef81e76b18f","year":2008},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:0ac18293d88bebbf7629854b271ad74f7c79a1b7d8c2995bb3c4bf0c1c10ec51","observation_id":"4420a238-25f7-4d3a-bfdd-c653ab3c5a0e","resolution":{"observed_at":"2026-07-12T07:28:35.163420Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-11T06:34:44.6726+00:00","source":"crossref"},{"observed_at":"2026-08-11T06:34:36.301508+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Systematic search for extreme and singular behaviour in some fundamental models of fluid mechanics","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:8a95edd447b451d651e2f0958751f713f32059e16c0f6e339c78c480f0a3b461","observation_id":"8485e43d-072f-4338-8fad-9fa0e39b67f7","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/j.jcp.2003.08","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:28:35.127741Z","title":"A computational framework for the regularization of adjoint analysis in multiscale pde systems","venue":null,"work_id":"ff553e4c-ce10-4758-a83f-a3f185f562e4","year":2004},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:248c228b6acfa87fd7d241d59fce75ca836aed1e999d053c1ab856ad06ce8b33","observation_id":"cd5d90da-e928-4021-a361-622e4dbfadd9","resolution":{"observed_at":"2026-07-12T07:28:35.129638Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-11T06:34:44.6726+00:00","source":"crossref"},{"observed_at":"2026-08-11T06:34:36.301508+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Systematic search for extreme behavior in 3D Navier- Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:a9b609fcfa6a5522d42e4364a3239d3603c8a63aaec6ae0797e59def72889d6e","observation_id":"c69a7b12-a744-4da5-b053-fb023ccff2e7","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2604.13338","last_updated":"2026-04-14T23:01:06Z","snapshot_observed_at":"2026-08-06T06:01:20.432841Z","submitted_at":"2026-04-14T23:01:06Z","title":"The Ladyzhenskaya-Prodi-Serrin Conditions and the Search for Extreme Behavior in 3D Navier-Stokes Flows","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2604.13338","snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"The ladyzhenskaya-prodi-serrin condi- tions and the search for extreme behavior in 3d navier-stokes flows","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"cited_paper":"/paper/2604.13338","citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:bde91bfc474aef1b5be4732d5a3608c9faaf4652f06317b314aae42c2293ffc0","observation_id":"eb1fb830-dc10-4b1a-908b-d391bd180896","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"A local smoothness criterion for solutions of the 3d navier-stokes equations","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:cd511f6e9d1afbc37d285f2f86295f9f75fabf52a8a8f58fcee985a6a2c41463","observation_id":"bd3bb036-5103-4774-8c2e-cecc5b93c5b1","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"On the interior regularity of weak solutions of the navier-stokes equations","venue":null,"work_id":null,"year":1962},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:e63a59d700e1923ace583c0639fd59de3bd9140a94afd6833cbc2f48580be933","observation_id":"56b78621-e512-42c0-8e01-16fd4fddaca4","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1908.04958","last_updated":"2020-07-10T04:41:18Z","snapshot_observed_at":"2026-07-06T08:14:16.969474Z","submitted_at":"2019-08-14T05:22:30Z","title":"Quantitative bounds for critically bounded solutions to the Navier-Stokes equations","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1908.04958","snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Quantitative bounds for critically bounded solutions to the navier-stokes equations.arXiv:1908.04958","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"cited_paper":"/paper/1908.04958","citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:abe7432f1c13b352ba72f9f6fb2f6a29d2cb0ad4eb0d553f191d6e85a280952f","observation_id":"ee4b5695-fe28-4b4a-a323-ca46881a1f83","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-12T07:22:30.053977Z","title":"Systematic search for singularities in 3d euler flows","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-07-12T07:22:30.053977Z"},"links":{"citing_paper":"/paper/2607.02739"},"observation_digest":"sha256:94ad5a4f8f673ebc0d5585b114af453c7c81770f5020b0656834fcec91cfa321","observation_id":"735198a0-9c20-4f4d-947b-a04babde7a86","resolution":{"observed_at":"2026-07-12T07:22:30.053977Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2607.02739","last_updated":"2026-07-02T20:20:01Z","latest_version":1,"primary_category":"math.AP","snapshot_observed_at":"2026-08-05T14:21:04.634754Z","submitted_at":"2026-07-02T20:20:01Z","title":"On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows"},"reference_resolution":{"displayed":32,"state_counts":{"malformed_identifier":3,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":23,"verified_exact":6,"verified_fuzzy":0},"total_outbound_references":32},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-11T06:34:44.6726+00:00","source":"crossref"},{"observed_at":"2026-08-11T06:34:36.301508+00:00","source":"retraction_watch"}],"thesis":"As of 11 August 2026, this Paper Citation Record lists 32 of 32 outbound references and 0 inbound Pith citation observations for arXiv:2607.02739."}