{"paper":{"title":"Spectral instability and non-uniqueness of mild solutions for the Keller-Segel system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The parabolic-elliptic Keller-Segel system is locally ill-posed in L^q(R^n) for n from 3 to 9 and q in the supercritical range [1, n/2).","cross_cats":["math.SP"],"primary_cat":"math.AP","authors_text":"Eliseo Luongo, Umberto Pappalettera","submitted_at":"2026-05-13T14:26:28Z","abstract_excerpt":"We show that the Cauchy problem associated with the parabolic-elliptic Keller-Segel model is locally ill-posed in $L^q(\\mathbb{R}^n)$ for dimensions $n \\in \\{3,\\dots,9\\}$ and throughout the supercritical range $q\\in [1,\\frac{n}{2})$. The non-uniqueness is driven by an instability mechanism in self-similarity variables, in the spirit of the program proposed by Jia and \\v{S}ver\\'ak for the three-dimensional Navier-Stokes equations."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that the Cauchy problem associated with the parabolic-elliptic Keller-Segel model is locally ill-posed in L^q(R^n) for dimensions n ∈ {3,…,9} and throughout the supercritical range q∈[1,n/2).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The non-uniqueness is driven by an instability mechanism in self-similarity variables, assuming the spectral instability from the Jia-Šverák Navier-Stokes program transfers directly to the Keller-Segel linearization without extra obstructions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The parabolic-elliptic Keller-Segel system is locally ill-posed in L^q(R^n) for n=3..9 and supercritical q in [1, n/2).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The parabolic-elliptic Keller-Segel system is locally ill-posed in L^q(R^n) for n from 3 to 9 and q in the supercritical range [1, n/2).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"31f7dda5ed02db7121927bf92f430ae961e8c356ce9955880a8ccb65fec597cf"},"source":{"id":"2605.13592","kind":"arxiv","version":1},"verdict":{"id":"8e263e9f-8146-46d9-bdf9-f6b35068d3c1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:17:51.884951Z","strongest_claim":"We show that the Cauchy problem associated with the parabolic-elliptic Keller-Segel model is locally ill-posed in L^q(R^n) for dimensions n ∈ {3,…,9} and throughout the supercritical range q∈[1,n/2).","one_line_summary":"The parabolic-elliptic Keller-Segel system is locally ill-posed in L^q(R^n) for n=3..9 and supercritical q in [1, n/2).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The non-uniqueness is driven by an instability mechanism in self-similarity variables, assuming the spectral instability from the Jia-Šverák Navier-Stokes program transfers directly to the Keller-Segel linearization without extra obstructions.","pith_extraction_headline":"The parabolic-elliptic Keller-Segel system is locally ill-posed in L^q(R^n) for n from 3 to 9 and q in the supercritical range [1, n/2)."},"references":{"count":46,"sample":[{"doi":"10.1007/s00205-016-1017-8","year":2016,"title":"Archive for Rational Mechanics and Analysis , VOLUME =","work_id":"fc647668-0c9d-416d-b3ec-d992c2243613","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.jfa.2024.110541","year":2024,"title":"Journal of Functional Analysis , VOLUME =","work_id":"1cd36c6c-66b5-4481-8957-9e9d8d92413a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s00039-025-00706-0","year":2025,"title":"Geometric and Functional Analysis , VOLUME =","work_id":"5bf8873e-b3b3-43ec-8b62-61b3ad6a39dc","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s40818-023-00155-8","year":2023,"title":"Annals of PDE , VOLUME =","work_id":"bcc3018c-158e-4e77-a93a-26d4b239ade3","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1515/crelle-2025-0025","year":2025,"title":"Journal f\\\"ur die Reine und Angewandte Mathematik","work_id":"1d9eea7a-4107-495d-8311-f6ea82d671ad","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":46,"snapshot_sha256":"990b1dc87bfab577032c00e396740ebe2f4fa2869685f36c560591bd46be9c29","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"}