{"paper":{"title":"Global existence and singularity formation for the generalized Constantin-Lax-Majda equation with dissipation: The real line vs. periodic domains","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["nlin.PS","nlin.SI"],"primary_cat":"math.AP","authors_text":"David M. Ambrose, Denis A. Silantyev, Michael Siegel, Pavel M. Lushnikov","submitted_at":"2022-07-15T15:42:46Z","abstract_excerpt":"The question of global existence versus finite-time singularity formation is considered for the generalized Constantin-Lax-Majda equation with dissipation $-\\Lambda^\\sigma$, where $\\widehat {{\\Lambda}^\\sigma}=|k|^\\sigma$, both for the problem on the circle $x \\in [-\\pi,\\pi]$ and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $\\sigma \\geq 1$ and all real values of an advection parameter $a$ when the data is small. We also derive new analytical solutions in both geometries when $a=0$, and on the real line when $a="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.07548","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2207.07548/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}