{"paper":{"title":"Energy and helicity conservation for the generalized quasi-geostrophic equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Huan Yu, Yanqing Wang, Yulin Ye","submitted_at":"2022-08-16T13:48:43Z","abstract_excerpt":"In this paper, we consider the 2-D generalized surface quasi-geostrophic equation with the velocity $v$ determined by $v=\\mathcal{R}^{\\perp}\\Lambda^{\\gamma-1}\\theta$. It is shown that the $L^p$ type energy norm of weak solutions is conserved provided $\\theta\\in L^{p+1}(0,T; {B}^{\\frac{\\gamma}{3}}_{p+1, c(\\mathbb{N})})$ for $0<\\gamma<\\frac32$ or $\\theta\\in L^{p+1}(0,T; {{B}}^{\\alpha}_{p+1,\\infty})~\\text{for any}~\\gamma-1<\\alpha<1 \\text{ with} ~\\frac{3}{2}\\leq \\gamma <2$. Moreover, we also prove that the helicity of weak solutions satisfying $\\nabla\\theta \\in L^{3}(0,T;\\dot{B}_{3,c(\\mathbb{N})}^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.07751","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/2208.07751/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"}