{"paper":{"title":"A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ademir Pastor, Felipe Linares, Luiz G. Farah","submitted_at":"2011-08-18T11:00:18Z","abstract_excerpt":"We consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+\\partial_x \\Delta u+\\partial_x(u^{k+1})=0$, where $k\\geq3$ is an integer number. For $k\\geq8$ we prove local well-posedness in the $L^2$-based Sobolev spaces $H^s(\\mathbb{R}^2)$, where $s$ is greater than the critical scaling index $s_k=1-2/k$. For $k\\geq 3$ we also establish a sharp criteria to obtain global $H^1(\\R^2)$ solutions. A nonlinear scattering result in $H^1(\\R^2)$ is also established assuming the initial data is small and belongs to a suitable Lebesgue space."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3714","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"}