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Such singular solutions are characterized by energy spectra that scale with the wave number $k$ as $k^{-2}$. ****\n  In this study, we carry out an analysis which verifies the dynamical features described above and derive upper bounds for $\\epsilon$ and $N$. It is found that $\\epsilon$ satisfies $\\epsilon \\le \\nu^{2\\alpha-1}\\norm{u_*}_\\infty^{2(1-\\alpha)} \\norm{(-\\Delta)^{\\alpha/2}u_*}^2$, where $\\alpha<1$ and $u_*=u("},"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":"0912.2007","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.flu-dyn","submitted_at":"2009-12-10T14:36:26Z","cross_cats_sorted":[],"title_canon_sha256":"968e7b5306a619c9a305c7f8f49e49bd885d73bd096acd9f6d24aa5458a0c384","abstract_canon_sha256":"887815bdf2fe7c17b1066fb9060cb827a87031df3ba8f4397d42377e3bc518a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:10:24.822676Z","signature_b64":"ktsr5r3Si13z4sl8nzwOAmGiXxgOus/VaGfXg/clV5MUwCf+U0S+AmrUm2poImnSxPEBbbZg5bM3OzCCOIgHBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"258f36e80c95f66404e5a15cd687defb8eb5abbe54674609be73b7ca64b43338","last_reissued_at":"2026-05-18T02:10:24.821886Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:10:24.821886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Energy dissipation and resolution of steep gradients in one-dimensional Burgers flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"physics.flu-dyn","authors_text":"Chuong V. 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