{"paper":{"title":"Well-posedness for a Family of Perturbations of the KDV Equation in Periodic Sobolev Spaces of Negative Order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ricardo Pastran, Xavier Carvajal","submitted_at":"2011-05-16T02:47:43Z","abstract_excerpt":"We establish local well-posedness in Sobolev spaces $H^s(\\mathbb{T})$, with $s\\geq -1/2$, for the initial value problem issues of the equation $$ u_t + u_{xxx}+\\eta Lu + uu_x=0;\\; x\\in \\mathbb{T},\\; t\\geq0, $$ where $\\eta >0$, $(Lu)^{\\wedge}(k)=-\\Phi(k)\\hat{u}(k)$, $k\\in \\mathbb{Z}$ and $\\Phi \\in \\mathbb{R}$ is bounded above. Particular cases of this problem are the Korteweg-de Vries-Burgers equation for $\\Phi(k)=-k^2$, the derivative Korteweg-de Vries-Kuramoto-Sivashinsky equation for $\\Phi(k)=k^2-k^4$, and the Ostrovsky-Stepanyams-Tsimring equation for $\\Phi(k)=|k|-|k|^3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2995","kind":"arxiv","version":3},"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"}