{"paper":{"title":"Norm inflation for quadratic derivative fractional nonlinear Schr\\\"odinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Quadratic derivative fractional nonlinear Schrödinger equations show norm inflation with infinite regularity loss below sharp exponents, implying ill-posedness.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mamoru Okamoto, Toshiki Kondo","submitted_at":"2026-01-28T06:26:46Z","abstract_excerpt":"We consider the Cauchy problem for quadratic derivative fractional nonlinear Schr\\\"odinger equations on $\\mathbb{R}$ or $\\mathbb{T}$. We determine the sharp exponents of the fractional derivatives for which the Cauchy problem is well-posed in the Sobolev space. Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms. By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For quadratic derivative fractional NLS, the Cauchy problem is ill-posed in Sobolev spaces below sharp fractional derivative exponents due to norm inflation with infinite loss of regularity.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Quadratic derivative fractional nonlinear Schrödinger equations show norm inflation with infinite regularity loss below sharp exponents, implying ill-posedness.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"135d6b1ce3363290dd445ee1052ff74259c6fbbff08aaa6152807d9739989173"},"source":{"id":"2601.20294","kind":"arxiv","version":2},"verdict":{"id":"8fae9b14-500c-4f6d-bbb6-bb18613bcf9b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T10:48:43.813609Z","strongest_claim":"By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness.","one_line_summary":"For quadratic derivative fractional NLS, the Cauchy problem is ill-posed in Sobolev spaces below sharp fractional derivative exponents due to norm inflation with infinite loss of regularity.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms.","pith_extraction_headline":"Quadratic derivative fractional nonlinear Schrödinger equations show norm inflation with infinite regularity loss below sharp exponents, implying ill-posedness."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.20294/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":26,"sample":[{"doi":"","year":2006,"title":"I. Bejenaru, T. Tao,Sharp well-posedness and ill-posedness results for a quadratic non-linear Schr¨ odinger equation, J. Funct. Anal.233(2006), no. 1, 228–259","work_id":"26b9d23b-bbc0-4756-acdc-d58071231afd","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"Chihara,The initial value problem for Schr¨ odinger equations on the torus, Int","work_id":"6c2e5641-1bd0-4ea2-b660-a7c0ed17bc0b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Christ,Illposedness of a Schr¨ odinger equation with derivative nonlinearity, preprint (https://math.berkeley.edu/~mchrist/preprints.html)","work_id":"a1d8954e-fbc7-4715-b247-028a471dc3d7","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"J. Chung, Z. Guo, S. Kwon, T. Oh,Normal form approach to global well-posedness of the quadratic derivative nonlinear Schr¨ odinger equation on the circle, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eai","work_id":"b72a5fa7-eb89-494a-a3f0-91611d6f462f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schroedinger equations","work_id":"a815334b-973b-438d-aef7-be35908ca093","ref_index":5,"cited_arxiv_id":"math/0006195","is_internal_anchor":true}],"resolved_work":26,"snapshot_sha256":"e7284ef28b61c645c7990c26556d6614261b802542300b5fedc167a665f2851f","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"33c2e04b8480d2fb0c972e085cc3027622cca6becb5aab8f494548915eb03c45"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}