{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:VEX4M32JAJSYFI5OF2ZWE55GAG","short_pith_number":"pith:VEX4M32J","schema_version":"1.0","canonical_sha256":"a92fc66f49026582a3ae2eb36277a60184fea7608b8a0c1bcca0d164b3aa94b3","source":{"kind":"arxiv","id":"1602.04533","version":1},"attestation_state":"computed","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}