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We prove the global in time existence of weak solution $u\\in L^q(0,\\infty; \\dot B^\\alpha_{pq}({\\mathbb R}^n_+))\\cap L^{q_0}(0, \\infty; L^{p_0}({\\mathbb R}^n_+)) $ for some $ 1<p_0, q_0<\\infty$ with $\\frac{n}{p_0} +\\frac2{q_0} =1$, when the given initial velocity has small norm in function space $ \\dot{B}_{p_0q_0,0}^{-\\fra"},"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":"1809.07025","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-19T06:40:49Z","cross_cats_sorted":[],"title_canon_sha256":"4cd81b0ecd870e942bf340498c1bd684c1a44c2d7d0fcae66e2b2f80dbeee725","abstract_canon_sha256":"5ecf8cd5f1406218804d7a4a48c98902dfcb4b24d2de1d986dada485d1019fb5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:09.567482Z","signature_b64":"Lc6KWOBItKvICFJzTYFQZk32qGulM9gW3bpij0A2KYT3eVaaxzQier8cFjUpYE/mGYbqyINo9z0SOVooQvnSBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b04ccd14510cf9e77230c6920df8263dba2033c0e241116937080ab45c7ff1f","last_reissued_at":"2026-05-17T23:56:09.566759Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:09.566759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global in time solvability of the Navier-Stokes equations in the half-space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bum Ja Jin, Tongkeun Chang","submitted_at":"2018-09-19T06:40:49Z","abstract_excerpt":"In this paper, we study the initial value problem of the Navier-Stokes equations in the half-space. 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