{"paper":{"title":"The local characterizations of the singularity formation for the MHD equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fan Wu, Wenke Tan","submitted_at":"2021-08-24T02:32:05Z","abstract_excerpt":"This paper characterizes the possible blow-up of solutions for the 3D magneto-hydrodynamics (MHD for short) equations. We first establish some $\\epsilon$-regularity criteria in $L^{q,\\infty}$ spaces for suitable weak solutions, and then together with an embedding theorem from $L^{p,\\infty}$ space into a Morrey type space to characterize the local behaviors of solutions near a potential singular point. More precisely, we show that if $z_{0}=\\left(t_{0}, x_{0}\\right)$ is a singular point, then for any $r>0$ it holds that $$ \\limsup _{t \\rightarrow t_{0}^{-}}\\left(\\left\\|u(t, x)-u(t)_{x_{0}, r}\\r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.10487","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.10487/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}