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That is, we prove that there exist two positive constants $c_0,C_0$ such that if \\begin{equation*}\n  \\|u_0^1+u^2_0,u^3_0\\|_{\\dot{B}_{p,1}^{-1+\\frac{3}{p}}} \\|u^1_0,u^2_0\\|_{\\dot{B}_{p,1}^{-1+\\frac{3}{p}}} \\exp\\{C_0 (\\|u_0\\|^{2}_{\\dot{B}_{\\infty,2}^{-1}}+\\|u_0\\|_{\\dot{B}_{\\infty,1}^{-1}})\\} \\leq c_0, \\end{equation*} then \\eqref{NS} has a unique global solution. As an application we construct two family of smooth solutions to the Navier-Stokes equations whose $\\B^"},"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":"1904.01779","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-03T05:45:29Z","cross_cats_sorted":[],"title_canon_sha256":"4a5a2b50f37b79b114c95c450a7f4b94bb5a257fd9d2d8c5d4797b81c4804156","abstract_canon_sha256":"897ffaee7f4456a6482ecfa64779b48fc17607c5ca2cd1c1d6a5f82f293d66a3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:14.284502Z","signature_b64":"roWvkMFmv94UUzgYKGVrV4UgAa686K7SsAz1fb+iojWXeahz5EIUuMuB+zMWo64fKJHyYMDyJrEIWmizyVqpCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4f6a2e749fc88018db49a0e274fe7af8ae3bea15d2d4c762ef05f7cffdadce1","last_reissued_at":"2026-05-17T23:49:14.283892Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:14.283892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Remarks on the global large solution to the three-dimensional incompressible Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jinlu Li, Yanghai Yu, Zhaoyang Yin","submitted_at":"2019-04-03T05:45:29Z","abstract_excerpt":"In this paper, we derive a new smallness hypothesis of initial data for the three-dimensional incompressible Navier-Stokes equations. That is, we prove that there exist two positive constants $c_0,C_0$ such that if \\begin{equation*}\n  \\|u_0^1+u^2_0,u^3_0\\|_{\\dot{B}_{p,1}^{-1+\\frac{3}{p}}} \\|u^1_0,u^2_0\\|_{\\dot{B}_{p,1}^{-1+\\frac{3}{p}}} \\exp\\{C_0 (\\|u_0\\|^{2}_{\\dot{B}_{\\infty,2}^{-1}}+\\|u_0\\|_{\\dot{B}_{\\infty,1}^{-1}})\\} \\leq c_0, \\end{equation*} then \\eqref{NS} has a unique global solution. 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