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The condition $a<0$ puts the model in an unstable phase-field regime: the nonlinear chemical potential may amplify, rather than restore, concentration flu"},"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":"2109.05299","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-09-11T15:20:42Z","cross_cats_sorted":[],"title_canon_sha256":"39011eaa76fa4a80fc58914873522bf2fc4762e19846643f836702cc4a453448","abstract_canon_sha256":"03a9f61503a598460a9fa8a2c7889c51df097337d047901b8dc421281d474619"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:03:39.638175Z","signature_b64":"7K1uD0CJN91ALzkeE+M92rJ9PYFt55KCecSZsVS7iWoGxQe4s1LI0zm8EzqfpmhZJXyb81SrFI5ARxCZe6fRBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"935b4fd491f2b888fcde27db88d96cf3d00f79ad2ec29debd75e0bd3aebc86ba","last_reissued_at":"2026-06-08T01:03:39.636998Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:03:39.636998Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global Existence for the unstable Cahn-Hilliard equation in 2D with a Shear Flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bingyang Hu, Dinghua Xu, Yeyu Zhang","submitted_at":"2021-09-11T15:20:42Z","abstract_excerpt":"In this paper, we study the advective unstable Cahn--Hilliard equation on $\\mathbb T^2$ with shear flow: \\begin{equation*}\n  \\begin{cases} u_t+Av_1(y) \\partial_x u+\\varepsilon \\Delta^2 u= \\Delta(a u^3+ b u^2) \\quad & \\quad \\textrm{on} \\quad \\mathbb T^2; \\\\ \\\\ u \\ \\textrm{periodic} \\quad & \\quad \\textrm{on} \\quad \\partial \\mathbb T^2, \\end{cases} \\end{equation*} where $u_0\\in H_0^2(\\mathbb T^2)$, $A,\\varepsilon>0$, $a<0$, and $b\\in\\mathbb R$. 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