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We first show a general criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case $\\alpha\\in]0,1[$ and $\\beta\\in ]\\alpha+1,2]$ we obtain the global well-posedness of the system with smooth initial data; a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1011.6214","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.AP","submitted_at":"2010-11-29T12:32:33Z","cross_cats_sorted":[],"title_canon_sha256":"f80ef2d18a3aeac637611f29fe52637ea3573392c3f9f41a0f9b2b6bb8d89d3f","abstract_canon_sha256":"96a8024d7023fac1cb536a278eeab1a9a63250f776bddb42bd309b09bff62e13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:55.125073Z","signature_b64":"UyDbFwMz5CyUD0yI+ImAcqk0YaBdb2cNZg+jY/SYxzHmPTZwX6wwflOSRW286+P7R0o8trYPlYyrlX4jAr5vDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb005803725728d40767a0b9682fa566398129cf086c4f5714975fc03202cd1f","last_reissued_at":"2026-05-18T04:14:55.124583Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:55.124583Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the regularity of a class of generalized quasi-geostrophic equations","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changxing Miao, Liutang Xue","submitted_at":"2010-11-29T12:32:33Z","abstract_excerpt":"In this article we consider the following generalized quasi-geostrophic equation\n  \\partial_t\\theta + u\\cdot\\nabla \\theta + \\nu \\Lambda^\\beta \\theta =0, \\quad u= \\Lambda^\\alpha \\mathcal{R}^\\bot\\theta, \\quad x\\in\\mathbb{R}^2, where $\\nu>0$, $\\Lambda:=\\sqrt{-\\Delta}$, $\\alpha\\in ]0,1[$ and $\\beta\\in ]0,2[$. 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