Paper Citation Record · LEDGER

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow

As of 19 August 2026, this Paper Citation Record lists 36 of 36 outbound references and 1 inbound Pith citation observation for arXiv:2506.10578.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2506.10578 v1

Coverage vector

measured 36 of 36 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T04:37:33.987295Z

measured 37 of 37 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-19T06:32:44.657259+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-06-26T20:34:21.556736Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: arxiv_reference, observed 2026-07-04T01:09:19.775444Z

Reference resolution

36 of 36 outbound references displayed

verified exact2
verified fuzzy24
unresolved10
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 680862f8-3f75-437e-8c6d-b21f43239fe4 · outbound

This paper cites and He S.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and He S

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.584635Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 1a2d3efc-789e-43b7-896a-95d17e904d6e · outbound

This paper cites an unresolved cited work.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-07T04:37:34.569642Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 42f025d9-a803-495b-b876-aa39593ae2a2 · outbound

This paper cites and Perthame B.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Perthame B

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.550073Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 105d7cf1-5de9-469f-99aa-68b7b72057d4 · outbound

This paper cites and Corrias L.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Corrias L

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.534251Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 785daa32-5be6-436c-9d9a-179b4cc3fccd · outbound

This paper cites and Zhang Z.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Zhang Z

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.516233Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation ea91d41b-43a5-4550-98ab-4d5b885bebca · outbound

This paper cites and Wang W.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Wang W

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.498591Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 2d7da5ce-69ab-4887-835e-2865a70f78a0 · outbound

This paper cites and Wang W.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Wang W

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.481648Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation fd41df37-7073-4d61-b0d0-f902833044ca · outbound

This paper cites Suppression of blow-up for the 3D Patlak-Keller-Segel-Navier-Stokes system via the Couette flow.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Suppression of blow-up for the 3D Patlak-Keller-Segel-Navier-Stokes system via the Couette flow

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-07T04:37:33.870388Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3d9ef1fc-13bb-4a43-a8e3-a0827bbfaf35 · outbound

This paper cites and Wei J.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Wei J

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.464856Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation f183e4ec-3474-4c02-be82-f1df605425c5 · outbound

This paper cites and Wang W.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Wang W

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.438268Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 80f80765-9df5-47b3-bf72-79d7567557c6 · outbound

This paper cites and Wang W.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Wang W

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.421652Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 1d37da94-caca-4670-8288-6d62c95016f0 · outbound

This paper cites an unresolved cited work.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-08-07T04:37:34.402860Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 4fbcaafa-a9d6-489c-bcb9-99566f7cacd9 · outbound

This paper cites an unresolved cited work.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Unresolved cited work

Reference 13

Resolution: unresolved; raw_fallback, observed 2026-08-07T04:37:34.381488Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 607ad01a-cf66-4006-a8f0-bb0fe62eed71 · outbound

This paper cites (2025) Time-dependent Flows and Their Applications in Parabolic-parabolic Patlak- Keller-Segel Systems Part I: Alternating Flows.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow (2025) Time-dependent Flows and Their Applications in Parabolic-parabolic Patlak- Keller-Segel Systems Part I: Alternating Flows

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.367566Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation d717d310-8b0f-4867-966e-c52cd84d67ed · outbound

This paper cites an unresolved cited work.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-07T04:37:34.353067Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 5eb7df8e-1bbc-4f6d-a1dc-a2705fd32cc7 · outbound

This paper cites and Tadmor E.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Tadmor E

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.340019Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 4728aed0-011b-40ec-b6d7-af0ee1a5d417 · outbound

This paper cites and Painter K.J.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Painter K.J

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.326497Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 71796bda-48c2-4240-a5c9-67f345cd98c7 · outbound

This paper cites (2023) Suppression of Chemotactic Singularity via Viscous Flow with Large Buoyancy.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow (2023) Suppression of Chemotactic Singularity via Viscous Flow with Large Buoyancy

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.312826Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation f1daddd9-a9d9-4876-937f-c533416857d0 · outbound

This paper cites and Kiselev A.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Kiselev A

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.299226Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 0128ae81-a2e8-4154-b7c1-2df6ca80a038 · outbound

This paper cites Suppression of chemotactic singularity by buoyancy.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Suppression of chemotactic singularity by buoyancy

Reference 20

Resolution: verified exact; local_arxiv, observed 2026-08-07T04:37:34.070764Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 746c9774-5ea6-489c-bd12-f4e10d0a3796 · outbound

This paper cites and Segel L.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Segel L

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.285526Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation d289c1b3-7fa5-4ec8-b0bb-529682439b60 · outbound

This paper cites and Xu X.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Xu X

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.272990Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 94160a2e-7446-4a11-9e83-14739a83e45a · outbound

This paper cites and Xu X.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Xu X

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.259123Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation ee027f47-6586-4ae1-aa82-4d3e6a8c87cf · outbound

This paper cites an unresolved cited work.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Unresolved cited work

Reference 24

Resolution: unresolved; raw_fallback, observed 2026-08-07T04:37:34.245959Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 9e3123a3-36d5-4027-87f3-6f11c458c302 · outbound

This paper cites an unresolved cited work.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Unresolved cited work

Reference 25

Resolution: unresolved; raw_fallback, observed 2026-08-07T04:37:34.232656Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation aeace536-2b86-45f9-b7cd-fde34310b614 · outbound

This paper cites Stable blow-up dynamic for the parabolic-parabolic Patlak-Keller-Segel model.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Stable blow-up dynamic for the parabolic-parabolic Patlak-Keller-Segel model

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-07T04:37:33.945172Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6f0839b5-419d-48fb-885c-c3da5176c51f · outbound

This paper cites and Wolansky G.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Wolansky G

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.218643Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation cc3b35ca-1bb6-4e24-8c7c-2a58ea79aaaf · outbound

This paper cites and Wang W.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Wang W

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.204488Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation a6dd8f6a-beff-4bb9-b50f-21cc99c83eb8 · outbound

This paper cites and Winkler M.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Winkler M

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.190606Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation b8736913-d01c-41fb-9474-b94ff23f2ccb · outbound

This paper cites and Winkler M.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Winkler M

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.175311Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 3fa85ec3-b2c1-4974-8eb7-db9bd9fd2158 · outbound

This paper cites and Winkler M.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Winkler M

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.159847Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 081ef4cd-5fa8-41ad-a361-c17e4aa3df91 · outbound

This paper cites Stability of a class of supercritical volume-filling chemotaxis-fluid model near Couette flow.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Stability of a class of supercritical volume-filling chemotaxis-fluid model near Couette flow

Reference 32

Resolution: verified exact; local_arxiv, observed 2026-08-07T04:37:34.033329Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 4f941f1c-d681-4a08-8979-fac024dff51d · outbound

This paper cites an unresolved cited work.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Unresolved cited work

Reference 33

Resolution: unresolved; raw_fallback, observed 2026-08-07T04:37:34.145493Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation fe9db2c1-187d-422d-94e2-fe336e632ef0 · outbound

This paper cites and Zhang Z.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Zhang Z

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.131198Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 6cd4d5eb-93d5-4192-9785-b14528283c7f · outbound

This paper cites an unresolved cited work.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow Unresolved cited work

Reference 35

Resolution: unresolved; raw_fallback, observed 2026-08-07T04:37:34.116976Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation e464433c-2ce9-40a2-b525-c73a8687804f · outbound

This paper cites and Zi R.

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and Zi R

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T04:37:34.102254Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Pith citing papers

Observation 9c87a8b5-84eb-4659-9688-88732b6f46df · inbound

Global bounded solutions for a class of generalized Hillen-Painter models near Couette flow in $\mathbb{R}^2$ cites this paper.

Global bounded solutions for a class of generalized Hillen-Painter models near Couette flow in $\mathbb{R}^2$ On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow

Reference 32

Resolution: verified exact; arxiv_id, observed 2026-07-04T01:09:19.776878Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.