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We prove that the bilinear estimate in $X_{s,b}$ with $s<\\frac{1}{2}$ is invalid.\n  We also prove that the problem is locally well-posed in $H^{s}(\\mathbf{T})$ with $\\frac{1}{6}<s<\\frac{1}{2}$ for small initial data. The result of this paper improves the result ","authors_text":"Wei Yan, Xiaoping Zhai, Yimin Zhang, Yongsheng Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-15T00:50:00Z","title":"The Cauchy problem for the shallow water typ equations in low regularity spaces on the circle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04533","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:dc7698911138792e050faeac0b97cd79c0283ed404ed17a3cec2b06d2dfa1b6a","target":"record","created_at":"2026-05-18T01:20:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"09c34ef77b5eb503ef8963a8b9477156ae7a57f845201ae30a3a7eb648f7be6d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-15T00:50:00Z","title_canon_sha256":"cdd2616169a2fedec8ce971f4054d42d58fe8e8cb8a33e3b1fa452fe214c2570"},"schema_version":"1.0","source":{"id":"1602.04533","kind":"arxiv","version":1}},"canonical_sha256":"a92fc66f49026582a3ae2eb36277a60184fea7608b8a0c1bcca0d164b3aa94b3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a92fc66f49026582a3ae2eb36277a60184fea7608b8a0c1bcca0d164b3aa94b3","first_computed_at":"2026-05-18T01:20:23.812610Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:23.812610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/pklHi1hhIMspU1NDRBS2jzuIAOw5isLRI1QJJ0nglmJcZBlVGOM6tY4OZ0KgQELmk7W0jMGd/p4n6Z9JRjfDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:23.813357Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.04533","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc7698911138792e050faeac0b97cd79c0283ed404ed17a3cec2b06d2dfa1b6a","sha256:e5718672d8bbc3e1efd601169c88c6a8e75d7db45ccef47ac7d24cedf200b8e2"],"state_sha256":"a24ff9575a01d2e27c4b92d5b050c4d3c9b3f7c49374bca7d9f7eb43f0c11d0e"}