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We prove that if $\\varepsilon \\ll \\nu^{1/2}$, where $\\nu$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\\varepsilon$-close in $H^1$ to $(e^{t \\nu \\partial_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t \\gg \\nu^{-1/3}$ by a mixing-enhanced dissipation effect, and experiences a transient grow"},"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":"1604.01831","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-06T23:32:33Z","cross_cats_sorted":["physics.flu-dyn"],"title_canon_sha256":"bcd211ba8c59d9aad629195278e03c8f97a335424344d3a6c283ba6b10c5a656","abstract_canon_sha256":"a8c5712c01d8b54cbeedcd22e31a20fe7b77d2b12c053897d3d7daf66d2c0b5b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:13.403692Z","signature_b64":"wRox2ueLkA1I93D5oT69Q3g5sX7YY/kAEDpsJpkLZ6skEzLlqOY7aTSphxN1wIABmhEymTrYlqS4Xl5++pYRAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64a8a8dfa5c6beaa3bdcab02c1b48716e6c31ca3f09cf6558d026dcd44688b04","last_reissued_at":"2026-05-18T01:04:13.403169Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:13.403169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Sobolev stability threshold for 2D shear flows near Couette","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["physics.flu-dyn"],"primary_cat":"math.AP","authors_text":"Fei Wang, Jacob Bedrossian, Vlad Vicol","submitted_at":"2016-04-06T23:32:33Z","abstract_excerpt":"We consider the 2D Navier-Stokes equation on $\\mathbb T \\times \\mathbb R$, with initial datum that is $\\varepsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\\| U(y) - y\\|_{H^{N+4}} \\ll 1$ and $N>1$. We prove that if $\\varepsilon \\ll \\nu^{1/2}$, where $\\nu$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\\varepsilon$-close in $H^1$ to $(e^{t \\nu \\partial_{yy}}U(y),0)$ for all $t>0$. 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