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We prove that the total vorticity will approach\n  \\begin{align*}\n  -1+\\frac{M_2(\\omega_{0})}{\\nu^{3/2}(1+t)^{5/2}} \\bar{\\Omega}\\left( \\frac{x}{\\sqrt{\\nu(1+t)^3}}, \\frac{y}{\\sqrt{\\nu(1+t)}} \\right),\n  \\end{align*}\n  where $-1$ is the vorticity of the Couette flow and $\\bar{\\Omega}$ is the kernel of a Fokker-Planck type operator $\\mathcal{L}=\\partial_Y^2+\\frac32 X\\partial_X+\\frac12 Y\\partial_Y+\\frac52-Y\\"},"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":"2605.21663","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-20T19:15:11Z","cross_cats_sorted":[],"title_canon_sha256":"2bff2520c4d10213fe8186d478f5fce27671c91ce1f038ab1ab1936686031619","abstract_canon_sha256":"19a991c827186db0f7c8b9da69d653673342363a191f50ff9f8a730c98b31f0a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:27.237827Z","signature_b64":"ksWPiZgaJrKEQmv2zc6GY2bSiljnaouzZGiU+IKLl2Sa8zk4//HP0HsSWA8LIqRen0yuPugWUtN+P54UvM/cBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"55b821a04f6716bf31e7b8b9078bc1c4a0a86d201392d6d347a129df4c03c0ec","last_reissued_at":"2026-05-22T01:03:27.237110Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:27.237110Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Long-time behaviour of two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nader Masmoudi, Ning Liu, Weiren Zhao","submitted_at":"2026-05-20T19:15:11Z","abstract_excerpt":"In this paper, we study the long-time behavior of solutions to the two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane with Navier-slip boundary conditions. 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