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This first enable us to construct weak solutions in $ L^2$ for this equation and to check that the \"solutions\" constructed by Kappeler and Topalov in the defocusing case satisfy the equation in some weak sense. In a second time, we prove that the solution-map associated with the mKdV and the KdV equation are discontinuous for the $ H^s(\\T) $ topology for respectively $ s<0$ and $ s<-1$. These last results are sharp."},"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":"1105.3601","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-05-18T12:23:47Z","cross_cats_sorted":[],"title_canon_sha256":"e042d390c537d23850acfa7f456b886e49ea9f81767a0da51aaef59b2f600dc1","abstract_canon_sha256":"9e3bb72b6ff43632a5c2a217fb8693408035a21f1c7c6045b1a0786df57bf1a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:13.327257Z","signature_b64":"r95ZtMRyv0pPc7V2mgfh9HaLempXTQ+Z6IxoRV7RTI8HMcRZStTq2ukcAzzU8X1NrkH2uuB+5GSpX4hgwbvdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b28c4641d40cd82ad56067d2f30e293625f2dde5c760a0a47b1699910471413c","last_reissued_at":"2026-05-18T04:17:13.326674Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:13.326674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp ill-posedness results for the KdV and mKdV equations on the torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Luc Molinet (LMPT)","submitted_at":"2011-05-18T12:23:47Z","abstract_excerpt":"We establish a new a priori bound for $ L^2 $-bounded sequences of solutions to the mKdV equations on the torus. 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