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A conformal mapping framework is set up to study this free boundary problem with $\\Omega_j$ being part of unknowns. For any given vorticities $\\mu_1, \\ldots, \\mu_N$ and small $r_1, \\ldots, r_N\\in \\mathbf{R}^+$, through a perturbation appr"},"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":"1809.06425","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-17T20:09:26Z","cross_cats_sorted":[],"title_canon_sha256":"00514c9ef3e0216e84d286619ccfde3e806f8ad61037e9b0a7019e66d8ce8dca","abstract_canon_sha256":"ba79fde27c6bd172910ad922cba7cb2108e86ec96c26f2fa5089de47bf43d336"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:52.209693Z","signature_b64":"CnD4Ju6Snha9cxc/MBtlN3dY5RK5acYgBdgWzlRDB5ni4IK82OQAv4JI3PWOhtwzPATnv9Ji4DOxK5i+6CBQAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c99dff0edb3a6ab7ccc96a1a6010c79209352939380e68ed90339822f1d8104","last_reissued_at":"2026-05-17T23:52:52.208989Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:52.208989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Concentrated steady vorticities of the Euler equation on 2-d domains and their linear stability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chongchun Zeng, Yiming Long, Yuchen Wang","submitted_at":"2018-09-17T20:09:26Z","abstract_excerpt":"We consider concentrated vorticities for the Euler equation on a smooth domain $\\Omega \\subset \\mathbf{R}^2$ in the form of \\[ \\omega = \\sum_{j=1}^N \\omega_j \\chi_{\\Omega_j}, \\quad |\\Omega_j| = \\pi r_j^2, \\quad \\int_{\\Omega_j} \\omega_j d\\mu =\\mu_j \\ne 0, \\] supported on well-separated vortical domains $\\Omega_j$, $j=1, \\ldots, N$, of small diameters $O(r_j)$. A conformal mapping framework is set up to study this free boundary problem with $\\Omega_j$ being part of unknowns. 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