{"paper":{"title":"Singularity formation for the incompressible Hall-MHD equations without resistivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dongho Chae, Shangkun Weng","submitted_at":"2013-12-19T12:51:06Z","abstract_excerpt":"In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space $H^{m}(\\mathbb{R}^3)$ for any $m>\\frac{7}{2}$. Namely, either the system is locally ill-posed in $H^{m}(\\mathbb{R}^3)$, or it is locally well-posed, but there exists an initial data in $H^{m}(\\mathbb{R}^3)$, for which the $H^{m}(\\mathbb{R}^3)$ norm of solution blows-up in finite time if $m>7/2$.\n  In the latter case we choose an axisymmetric initial data $u_0(x)=u_{0r}(r,z)e_r+ b_{0z}(r,z)e_z$ and $B_0(x)=b_{0\\theta}(r,z)e_{\\theta}$, and reduce the system to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5519","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"}