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The zeroth order term takes the form of local Maxwellian: $ F_{0}(t,x,v)=\\frac{\\rho_{0}(t,x)}{(2\\pi \\theta_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2\\theta_{0}(t,x)},\\text{\\ }\\theta_{0}(t,x)=K\\rho_{0}^{2/3}(t,x).$ Our main result states that if the Hilbert expansion is valid at $t=0$ for well-prepared small initial data with irrotational velocity $u_0$, then it is valid for $0\\leq t\\leq \\varepsilon ^{-{1/2}\\frac{2k-3}{2k-2}},$ where $\\rho_{0}(t,x)$ and $ u_{0}(t,x)$"},"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":"0910.5512","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-10-28T22:38:47Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"ad79345de65e12bb78d3466187f13d7d617bef070895841982964687fe07e787","abstract_canon_sha256":"a5ecbb0a7ffdf0a61ea7f0939d1b7bc0bc57fedef433a51015c5b949b82b2dd7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:11:01.943859Z","signature_b64":"E67EVkM5BK+C5kZhRIs/kmQmhitHe3o0N7NJ5MUxYi8EwJ1WE/QxeaO0ln5jCrNF8SJZ+CJQPAUDsARX2X6mAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8991347ae8964e1ef482d91342033827be8eeefef0cac32cd8697c9fbc3fad89","last_reissued_at":"2026-05-18T02:11:01.943084Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:11:01.943084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Juhi Jang, Yan Guo","submitted_at":"2009-10-28T22:38:47Z","abstract_excerpt":"We study the Hilbert expansion for small Knudsen number $\\varepsilon$ for the Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term takes the form of local Maxwellian: $ F_{0}(t,x,v)=\\frac{\\rho_{0}(t,x)}{(2\\pi \\theta_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2\\theta_{0}(t,x)},\\text{\\ }\\theta_{0}(t,x)=K\\rho_{0}^{2/3}(t,x).$ Our main result states that if the Hilbert expansion is valid at $t=0$ for well-prepared small initial data with irrotational velocity $u_0$, then it is valid for $0\\leq t\\leq \\varepsilon ^{-{1/2}\\frac{2k-3}{2k-2}},$ where $\\rho_{0}(t,x)$ and $ u_{0}(t,x)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.5512","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":"0910.5512","created_at":"2026-05-18T02:11:01.943212+00:00"},{"alias_kind":"arxiv_version","alias_value":"0910.5512v1","created_at":"2026-05-18T02:11:01.943212+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.5512","created_at":"2026-05-18T02:11:01.943212+00:00"},{"alias_kind":"pith_short_12","alias_value":"RGITI6XISZHB","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_16","alias_value":"RGITI6XISZHB55EC","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_8","alias_value":"RGITI6XI","created_at":"2026-05-18T12:26:01.383474+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/RGITI6XISZHB55EC3EJUEAZYE6","json":"https://pith.science/pith/RGITI6XISZHB55EC3EJUEAZYE6.json","graph_json":"https://pith.science/api/pith-number/RGITI6XISZHB55EC3EJUEAZYE6/graph.json","events_json":"https://pith.science/api/pith-number/RGITI6XISZHB55EC3EJUEAZYE6/events.json","paper":"https://pith.science/paper/RGITI6XI"},"agent_actions":{"view_html":"https://pith.science/pith/RGITI6XISZHB55EC3EJUEAZYE6","download_json":"https://pith.science/pith/RGITI6XISZHB55EC3EJUEAZYE6.json","view_paper":"https://pith.science/paper/RGITI6XI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0910.5512&json=true","fetch_graph":"https://pith.science/api/pith-number/RGITI6XISZHB55EC3EJUEAZYE6/graph.json","fetch_events":"https://pith.science/api/pith-number/RGITI6XISZHB55EC3EJUEAZYE6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RGITI6XISZHB55EC3EJUEAZYE6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RGITI6XISZHB55EC3EJUEAZYE6/action/storage_attestation","attest_author":"https://pith.science/pith/RGITI6XISZHB55EC3EJUEAZYE6/action/author_attestation","sign_citation":"https://pith.science/pith/RGITI6XISZHB55EC3EJUEAZYE6/action/citation_signature","submit_replication":"https://pith.science/pith/RGITI6XISZHB55EC3EJUEAZYE6/action/replication_record"}},"created_at":"2026-05-18T02:11:01.943212+00:00","updated_at":"2026-05-18T02:11:01.943212+00:00"}