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When $\\alpha\\in]0,1[$, the equation was firstly introduced by Constantin, Iyer and Wu in \\cite{ref ConstanIW}. Here, by using the modulus of continuity method, we prove the global well-posedness of the system with the smooth initial data. As a byproduct, we also show that for every $\\alpha\\in ]0,2[$, the Lipschitz norm of the solutio"},"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":"0901.1368","kind":"arxiv","version":6},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AP","submitted_at":"2009-01-10T10:34:56Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"9b3578feaed151efb87dd7bb0444c6eda9ca28ce1cd49366219617281fc44b8c","abstract_canon_sha256":"2228dc66ac9b783248587c4cf55846bfd158538a2aa5fb7d270bfe2d04ef564b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:06.222494Z","signature_b64":"anXymYXN8qHZ1MA+v3BtVr1ApVTHmNdHaFFM1OsSTQzNa/GEbtV/id8u07tIUxzokmPwp6mhcUqCrK01kiwRDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4bd5fb1475b8baf7c653c52ff7442ad14911183a41100e0912b06db26c72d82f","last_reissued_at":"2026-05-18T04:11:06.221834Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:06.221834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global Wellposedness for a Modified Critical Dissipative Quasi-Geostrophic Equation","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Changxing Miao, Liutang Xue","submitted_at":"2009-01-10T10:34:56Z","abstract_excerpt":"In this paper we consider the following modified quasi-geostrophic equation \n  \\partial_{t}\\theta+u\\cdot\\nabla\\theta+\\nu |D|^{\\alpha}\\theta=0,\n  \\quad u=|D|^{\\alpha-1}\\mathcal{R}^{\\bot}\\theta,\\quad x\\in\\mathbb{R}^2 with $\\nu>0$ and $\\alpha\\in ]0,1[\\,\\cup \\,]1,2[$. When $\\alpha\\in]0,1[$, the equation was firstly introduced by Constantin, Iyer and Wu in \\cite{ref ConstanIW}. Here, by using the modulus of continuity method, we prove the global well-posedness of the system with the smooth initial data. 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