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Then we have Theorem: If $D_0$ is a $C^{1 + \\gamma}$ domain, then the initial value problem above has a solution g"},"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":"1507.07831","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-28T16:25:18Z","cross_cats_sorted":[],"title_canon_sha256":"94a787defa7f318e47e649d0b5ab9a756a22effc8674a98c6d0476db062aaa50","abstract_canon_sha256":"834e8991456b5e59b6c59962675a8f81d8b79df767d33e04391c85c987d4bff4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:52.437243Z","signature_b64":"qvAbYBNY3P5+Nrl9+0qc2nlX9lfrxSG0iyyxSNIevFMMzZU03tQM0PQwrqSRFx52W9FKRvAqSneAwMP6W9ZMCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b3dd5a2f3dce6446a5b9ee0940496a2404e25816cec5c8322853b4365805267","last_reissued_at":"2026-05-18T01:07:52.436806Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:52.436806Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The regularity of the boundary of a multidimensional aggregation patch","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrea Bertozzi, Joan Verdera, John Garnett, Thomas Laurent","submitted_at":"2015-07-28T16:25:18Z","abstract_excerpt":"Let $d \\geq 2$ and let $N(y)$ be the fundamental solution of the Laplace equation in $R^d$ We consider the aggregation equation $$ \\frac{\\partial \\rho}{\\partial t} + \\operatorname{div}(\\rho v) =0, v = -\\nabla N * \\rho$$ with initial data $\\rho(x,0) = \\chi_{D_0}$, where $\\chi_{D_0}$ is the indicator function of a bounded domain $D_0 \\subset R^d.$ We now fix $0 < \\gamma < 1$ and take $D_0$ to be a bounded $C^{1+\\gamma}$ domain (a domain with smooth boundary of class $C^{1+\\gamma}$). Then we have Theorem: If $D_0$ is a $C^{1 + \\gamma}$ domain, then the initial value problem above has a solution g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07831","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1507.07831","created_at":"2026-05-18T01:07:52.436876+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.07831v2","created_at":"2026-05-18T01:07:52.436876+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07831","created_at":"2026-05-18T01:07:52.436876+00:00"},{"alias_kind":"pith_short_12","alias_value":"JM65LIXT3TTE","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"JM65LIXT3TTEI2S3","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"JM65LIXT","created_at":"2026-05-18T12:29:27.538025+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JM65LIXT3TTEI2S3T3QJIBEWUJ","json":"https://pith.science/pith/JM65LIXT3TTEI2S3T3QJIBEWUJ.json","graph_json":"https://pith.science/api/pith-number/JM65LIXT3TTEI2S3T3QJIBEWUJ/graph.json","events_json":"https://pith.science/api/pith-number/JM65LIXT3TTEI2S3T3QJIBEWUJ/events.json","paper":"https://pith.science/paper/JM65LIXT"},"agent_actions":{"view_html":"https://pith.science/pith/JM65LIXT3TTEI2S3T3QJIBEWUJ","download_json":"https://pith.science/pith/JM65LIXT3TTEI2S3T3QJIBEWUJ.json","view_paper":"https://pith.science/paper/JM65LIXT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.07831&json=true","fetch_graph":"https://pith.science/api/pith-number/JM65LIXT3TTEI2S3T3QJIBEWUJ/graph.json","fetch_events":"https://pith.science/api/pith-number/JM65LIXT3TTEI2S3T3QJIBEWUJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JM65LIXT3TTEI2S3T3QJIBEWUJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JM65LIXT3TTEI2S3T3QJIBEWUJ/action/storage_attestation","attest_author":"https://pith.science/pith/JM65LIXT3TTEI2S3T3QJIBEWUJ/action/author_attestation","sign_citation":"https://pith.science/pith/JM65LIXT3TTEI2S3T3QJIBEWUJ/action/citation_signature","submit_replication":"https://pith.science/pith/JM65LIXT3TTEI2S3T3QJIBEWUJ/action/replication_record"}},"created_at":"2026-05-18T01:07:52.436876+00:00","updated_at":"2026-05-18T01:07:52.436876+00:00"}