{"paper":{"title":"On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Omar Lazar","submitted_at":"2016-03-16T13:45:08Z","abstract_excerpt":"We study a 1D transport equation with nonlocal velocity with subcritical or supercritical dissipation. For all data in the weighted Sobolev space $H^{k}(w_{\\lambda,\\kappa}) \\cap L^{\\infty},$ with $k=\\max(0,3/2-\\alpha)$ and $w_{\\lambda, \\kappa}$ is a given family of Muckenhoupt weights. We prove a global existence result in the subcritical case $\\alpha \\in (1,2)$. We also prove a local existence theorem for large data in $H^{2}(w_{\\lambda, \\kappa})\\cap L^{\\infty}$ in the supercritical case $\\alpha \\in (0,1)$. The proofs are based on the use of the weighted Littlewood-Paley theory, interpolation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.05096","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"}