{"paper":{"title":"The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Eddye Bustamante, Jorge Mej\\'ia, Jos\\'e Jim\\'enez Urrea","submitted_at":"2017-10-23T16:54:02Z","abstract_excerpt":"In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equation $$\\left. \\begin{array}{rl} u_t+D_x^{\\alpha} u_x +\\mathcal Hu_{yy} +uu_x &=0,\\qquad\\qquad (x,y)\\in\\mathbb R^2,\\; t\\in\\mathbb R, u(x,y,0)&=u_0(x,y), \\end{array} \\right\\}\\,,$$ where $0<\\alpha\\leq1$, $D_x^{\\alpha}$ denotes the operator defined through the Fourier transform by \\begin{align} (D_x^{\\alpha}f)\\widehat{\\;}(\\xi,\\eta):=|\\xi|^{\\alpha}\\widehat{f}(\\xi,\\eta)\\,, \\end{align} and $\\mathcal H$ denotes the Hilbert transform with respect to the variable $x$, is locally well "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.08380","kind":"arxiv","version":2},"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"}