{"paper":{"title":"Weighted fractional chain rule and nonlinear wave equations with minimal regularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Chengbo Wang, Jin-Cheng Jiang, Kunio Hidano, Sanghyuk Lee","submitted_at":"2016-05-22T07:02:40Z","abstract_excerpt":"We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data: $\\Box u = a |\\partial_t u|^2+b|\\nabla_x u|^2$, $u(0,x)=u_0(x)\\in H^{s}_{\\mathrm{rad}}$, $\\partial_t u(0,x)=u_1(x)\\in H^{s-1}_{\\mathrm{rad}}$. It has been known that the problem is well-posed for $s\\ge 2$ and ill-posed for $s<3/2$. In this paper, we prove unconditional well-posedness up to the scaling invariant regularity, that is to say, for $s>3/2$ and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06748","kind":"arxiv","version":3},"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"}