{"paper":{"title":"The Cauchy problem for the shallow water typ equations in low regularity spaces on the circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wei Yan, Xiaoping Zhai, Yimin Zhang, Yongsheng Li","submitted_at":"2016-02-15T00:50:00Z","abstract_excerpt":"In this paper, we investigate the Cauchy problem for the shallow water type equation \n\\[ u_{t}+\\partial_{x}^{3}u\n  + \\frac{1}{2}\\partial_{x}(u^{2})+\\partial_{x}\n  (1-\\partial_{x}^{2})^{-1}\\left[u^{2}+\\frac{1}{2}u_{x}^{2}\\right]=0,x\\in {\\mathbf T}=\\R/2\\pi\n  \\lambda \\] with low regularity data in the periodic settings and $\\lambda\\geq1$. 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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04533","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}