{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:RZYXLT7VARZ2AGS55TVGUHERJN","short_pith_number":"pith:RZYXLT7V","schema_version":"1.0","canonical_sha256":"8e7175cff50473a01a5decea6a1c914b5a74e4df453cbcc0374cd95a3bd94c9e","source":{"kind":"arxiv","id":"0808.2324","version":1},"attestation_state":"computed","paper":{"title":"On Stability of Pseudo-Conformal Blowup for L^2-critical Hartree NLS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Enno Lenzmann, Joachim Krieger, Pierre Raphael","submitted_at":"2008-08-18T00:39:50Z","abstract_excerpt":"We consider $L^2$-critical focusing nonlinear Schroedinger equations with Hartree type nonlinearity $$i \\pr_t u = -\\DD u - \\big (\\Phi \\ast |u|^2 \\big) u \\quad {in $\\RR^4$},$$ where $\\Phi(x)$ is a perturbation of the convolution kernel $|x|^{-2}$. Despite the lack of pseudo conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions $u(t,x)$ that exhibit the pseudo-conformal blowup rate $$ \\| \\nabla u(t) \\|_{L^2_x} \\sim \\frac{1}{|t|} \\quad {as} \\quad t \\nearrow 0 . $$ Furthermore, we prove the finite-codimensional stability of this conformal blow"},"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":"0808.2324","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-08-18T00:39:50Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"4306ce3bf34b53609d74e0e64a84f06a660a8277f813c26adfda2da5f797d8c0","abstract_canon_sha256":"901705bf961323b1d43d614cbe7e0e845071742c09ce7dc1a088abef3140ec8e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:29.490160Z","signature_b64":"+sNhSRDbiqkvr9m7mJLN9XDbdsinuDZdWpwK+yqlQwW7QBar+vPdRT6zSBNv0u5Gj+wh+PqS41AcOYXsbmN8Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e7175cff50473a01a5decea6a1c914b5a74e4df453cbcc0374cd95a3bd94c9e","last_reissued_at":"2026-05-18T04:07:29.489624Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:29.489624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Stability of Pseudo-Conformal Blowup for L^2-critical Hartree NLS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Enno Lenzmann, Joachim Krieger, Pierre Raphael","submitted_at":"2008-08-18T00:39:50Z","abstract_excerpt":"We consider $L^2$-critical focusing nonlinear Schroedinger equations with Hartree type nonlinearity $$i \\pr_t u = -\\DD u - \\big (\\Phi \\ast |u|^2 \\big) u \\quad {in $\\RR^4$},$$ where $\\Phi(x)$ is a perturbation of the convolution kernel $|x|^{-2}$. Despite the lack of pseudo conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions $u(t,x)$ that exhibit the pseudo-conformal blowup rate $$ \\| \\nabla u(t) \\|_{L^2_x} \\sim \\frac{1}{|t|} \\quad {as} \\quad t \\nearrow 0 . $$ Furthermore, we prove the finite-codimensional stability of this conformal blow"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.2324","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":"0808.2324","created_at":"2026-05-18T04:07:29.489717+00:00"},{"alias_kind":"arxiv_version","alias_value":"0808.2324v1","created_at":"2026-05-18T04:07:29.489717+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0808.2324","created_at":"2026-05-18T04:07:29.489717+00:00"},{"alias_kind":"pith_short_12","alias_value":"RZYXLT7VARZ2","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_16","alias_value":"RZYXLT7VARZ2AGS5","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_8","alias_value":"RZYXLT7V","created_at":"2026-05-18T12:25:58.018023+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/RZYXLT7VARZ2AGS55TVGUHERJN","json":"https://pith.science/pith/RZYXLT7VARZ2AGS55TVGUHERJN.json","graph_json":"https://pith.science/api/pith-number/RZYXLT7VARZ2AGS55TVGUHERJN/graph.json","events_json":"https://pith.science/api/pith-number/RZYXLT7VARZ2AGS55TVGUHERJN/events.json","paper":"https://pith.science/paper/RZYXLT7V"},"agent_actions":{"view_html":"https://pith.science/pith/RZYXLT7VARZ2AGS55TVGUHERJN","download_json":"https://pith.science/pith/RZYXLT7VARZ2AGS55TVGUHERJN.json","view_paper":"https://pith.science/paper/RZYXLT7V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0808.2324&json=true","fetch_graph":"https://pith.science/api/pith-number/RZYXLT7VARZ2AGS55TVGUHERJN/graph.json","fetch_events":"https://pith.science/api/pith-number/RZYXLT7VARZ2AGS55TVGUHERJN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RZYXLT7VARZ2AGS55TVGUHERJN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RZYXLT7VARZ2AGS55TVGUHERJN/action/storage_attestation","attest_author":"https://pith.science/pith/RZYXLT7VARZ2AGS55TVGUHERJN/action/author_attestation","sign_citation":"https://pith.science/pith/RZYXLT7VARZ2AGS55TVGUHERJN/action/citation_signature","submit_replication":"https://pith.science/pith/RZYXLT7VARZ2AGS55TVGUHERJN/action/replication_record"}},"created_at":"2026-05-18T04:07:29.489717+00:00","updated_at":"2026-05-18T04:07:29.489717+00:00"}