{"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. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05540","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"}