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Our argument also shows that the compressible viscous heat-conductive flows is ill-posed for the initial density, velocity and temperature belonging to the critical Besov space $(\\dot{B}^{\\f 3p}_{p,1}+\\bar{\\rho},\\,\\dot{B}^{\\f 3p-1}_{p,1},\\,\\dot{B}^{\\f 3p-2}_{p,1})$ for $p>3$."},"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":"1109.6092","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-09-28T04:12:54Z","cross_cats_sorted":[],"title_canon_sha256":"eca5ef88753e0614aa743076b45401aaa2bb1996cdbe2ca46a0fcab1a9945b40","abstract_canon_sha256":"89a049f8675d7860bdbffd54e4098ac689b821375ba2263a77d30ed4bf693b3f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:31.073890Z","signature_b64":"IRSOjhZsl+txR6LFstC0qCWqLTmn7S7l1wEV1jASBoS71vBJIz6xhlzwg5P02jIN0LLWA7wZ7vfC+2TEAenZBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c2dcda50985b185f88b0be516789230f594cf9a3afcf52a4e9314841f1bf233","last_reissued_at":"2026-05-18T01:24:31.073215Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:31.073215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changxing Miao, Qionglei Chen, Zhifei Zhang","submitted_at":"2011-09-28T04:12:54Z","abstract_excerpt":"We prove the ill-posedness of the 3-D baratropic Navier-Stokes equation for the initial density and velocity belonging to the critical Besov space $(\\dot{B}^{\\f 3p}_{p,1}+\\bar{\\rho},\\,\\dot{B}^{\\f 3p-1}_{p,1})$ for $p>6$ in the sense that a ``norm inflation\" happens in finite time, here $\\bar{\\rho}$ is a positive constant. 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