{"paper":{"title":"Global existence of weak solutions to dissipative transport equations with nonlocal velocity","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Hantaek Bae, Omar Lazar, Rafael Granero-Belinch\\'on","submitted_at":"2016-09-14T17:42:43Z","abstract_excerpt":"We consider 1D dissipative transport equations with nonlocal velocity field: \\[ \\theta_t+u\\theta_x+\\delta u_{x} \\theta+\\Lambda^{\\gamma}\\theta=0, \\quad u=\\mathcal{N}(\\theta), \\] where $\\mathcal{N}$ is a nonlocal operator given by a Fourier multiplier. Especially we consider two types of nonlocal operators:\n  $\\mathcal{N}=\\mathcal{H}$, the Hilbert transform,\n  $\\mathcal{N}=(1-\\partial_{xx} )^{-\\alpha}$.\n  In this paper, we show several global existence of weak solutions depending on the range of $\\gamma$ and $\\delta$. When $0<\\gamma<1$, we take initial data having finite energy, while we take in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04357","kind":"arxiv","version":4},"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"}