{"paper":{"title":"On a transport equation with nonlocal drift","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Luis Silvestre, Vlad Vicol","submitted_at":"2014-08-05T18:23:12Z","abstract_excerpt":"In \\cite{CordobaCordobaFontelos05}, C\\'ordoba, C\\'ordoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions \\[ \\partial_t \\theta +u \\; \\partial_x \\theta = 0, \\qquad u = H \\theta, \\] where $H$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible H\\\"older regularization effects of this equation and its consequences to the equation with diffusion \\[ \\partial_t \\theta + u \\; \\partial_x \\theta + \\Lambda^\\gamma \\theta = 0, \\qquad u = H \\th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.1056","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"}