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In this paper, we construct the exact solutions in non-radial symmetry to the pressureless Euler equations in $R^{N}:$%  [c]{c}% \\rho(t,\\vec{x})=\\frac{f(\\frac{1}{a(t)^{s}}\\underset{i=1}{\\overset {N}{\\sum}}x_{i}^{s})}{a(t)^{N}}\\text{,}\\vec{u}(t,\\vec{x}% )=\\frac{\\overset{\\cdot}{a}(t)}{a(t)}\\vec{x}, a(t)=a_{1}+a_{2}t. \\label{eq234}%  where the arbitrary function $f\\geq0$ and $f\\in C^{1};$ $s\\geq1$, $a_{1}>0$ and $a_{2}$ are constants$.$\\newline In particular, for $a_{2}<0$, the solutions blow up on the f"},"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":"0910.1272","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"astro-ph.SR","submitted_at":"2009-10-07T14:23:09Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"6dfe9c2896017c8a0f7d7b8cbf48d0aec324196e4a1966eaba752ec9b610bc61","abstract_canon_sha256":"c17078a5549a61f339fc18f6aa2903abecb5158613ed54f576787c56f44d73b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:31.955150Z","signature_b64":"C27jPTZX3kBQ6M/jK6lsHM784RspK4YrLa54UwOaRjOsURyiV/jscBtC5tIfftsz7AUXqMSFVaQ+WNZbGxFSBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"496c6f2fbc9ba6d7d1e82f14dba5ad172d06f64c2985de6569770aeb65764063","last_reissued_at":"2026-05-18T04:26:31.954575Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:31.954575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"astro-ph.SR","authors_text":"Manwai Yuen","submitted_at":"2009-10-07T14:23:09Z","abstract_excerpt":"The pressureless Euler equations can be used as simple models of cosmology or plasma physics. In this paper, we construct the exact solutions in non-radial symmetry to the pressureless Euler equations in $R^{N}:$%  [c]{c}% \\rho(t,\\vec{x})=\\frac{f(\\frac{1}{a(t)^{s}}\\underset{i=1}{\\overset {N}{\\sum}}x_{i}^{s})}{a(t)^{N}}\\text{,}\\vec{u}(t,\\vec{x}% )=\\frac{\\overset{\\cdot}{a}(t)}{a(t)}\\vec{x}, a(t)=a_{1}+a_{2}t. \\label{eq234}%  where the arbitrary function $f\\geq0$ and $f\\in C^{1};$ $s\\geq1$, $a_{1}>0$ and $a_{2}$ are constants$.$\\newline In particular, for $a_{2}<0$, the solutions blow up on the f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.1272","kind":"arxiv","version":1},"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":"0910.1272","created_at":"2026-05-18T04:26:31.954675+00:00"},{"alias_kind":"arxiv_version","alias_value":"0910.1272v1","created_at":"2026-05-18T04:26:31.954675+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.1272","created_at":"2026-05-18T04:26:31.954675+00:00"},{"alias_kind":"pith_short_12","alias_value":"JFWG6L54TOTN","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_16","alias_value":"JFWG6L54TOTNPUPI","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_8","alias_value":"JFWG6L54","created_at":"2026-05-18T12:26:00.592388+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/JFWG6L54TOTNPUPIF4KNXJNNC4","json":"https://pith.science/pith/JFWG6L54TOTNPUPIF4KNXJNNC4.json","graph_json":"https://pith.science/api/pith-number/JFWG6L54TOTNPUPIF4KNXJNNC4/graph.json","events_json":"https://pith.science/api/pith-number/JFWG6L54TOTNPUPIF4KNXJNNC4/events.json","paper":"https://pith.science/paper/JFWG6L54"},"agent_actions":{"view_html":"https://pith.science/pith/JFWG6L54TOTNPUPIF4KNXJNNC4","download_json":"https://pith.science/pith/JFWG6L54TOTNPUPIF4KNXJNNC4.json","view_paper":"https://pith.science/paper/JFWG6L54","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0910.1272&json=true","fetch_graph":"https://pith.science/api/pith-number/JFWG6L54TOTNPUPIF4KNXJNNC4/graph.json","fetch_events":"https://pith.science/api/pith-number/JFWG6L54TOTNPUPIF4KNXJNNC4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JFWG6L54TOTNPUPIF4KNXJNNC4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JFWG6L54TOTNPUPIF4KNXJNNC4/action/storage_attestation","attest_author":"https://pith.science/pith/JFWG6L54TOTNPUPIF4KNXJNNC4/action/author_attestation","sign_citation":"https://pith.science/pith/JFWG6L54TOTNPUPIF4KNXJNNC4/action/citation_signature","submit_replication":"https://pith.science/pith/JFWG6L54TOTNPUPIF4KNXJNNC4/action/replication_record"}},"created_at":"2026-05-18T04:26:31.954675+00:00","updated_at":"2026-05-18T04:26:31.954675+00:00"}