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We show that for sufficiently small $u_0 \\in \\dot B^1_{\\infty,1}(\\mathbb R^N)$, there exists a global-in-time mild solution. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel."},"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":"1509.05540","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-09-18T08:24:07Z","cross_cats_sorted":[],"title_canon_sha256":"160e7869e36708c301314a78a77932a643376223e4abe68c2532aba8a27016dc","abstract_canon_sha256":"2c09ebe92e7653e702641944b7334b1f0d4868b4d39578d89719682593408b10"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.888571Z","signature_b64":"cI2Ro3Z1xieyI6WNidZwBFQsRrlB/zCqujm9Glrv7Ixu9WZDV7QztzNg9VxOgNuEP+5YZU7WB3ME3eYrUqJ2Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8474a54b06462a24e037640eb95b69dd4a338e62e551c97407753c22c5502f1","last_reissued_at":"2026-05-18T01:32:42.888133Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.888133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence of mild solutions for the Hamilton-Jacobi equation with critical fractional viscosity in the Besov spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tatsuki Kawakami, Tsukasa Iwabuchi","submitted_at":"2015-09-18T08:24:07Z","abstract_excerpt":"We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $$ \\partial_t u + (-\\Delta)^{ 1/2} u = |\\nabla u|^p, \\quad x \\in \\mathbb R^N, t > 0, \\qquad u(x,0) = u_0(x) , \\quad x \\in \\mathbb R^N, $$ where $p > 1$ and $u_0 \\in B^1_{r,1}(\\mathbb R^N) \\cap B^1_{\\infty,1} (\\mathbb R^N)$ with $r \\in [1,\\infty]$. We show that for sufficiently small $u_0 \\in \\dot B^1_{\\infty,1}(\\mathbb R^N)$, there exists a global-in-time mild solution. 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