{"paper":{"title":"Ill-posedness for the 3D inhomogeneous Navier-Stokes equations in the critical Besov space near $L^6$ framework","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Renhui Wan","submitted_at":"2016-09-15T09:42:30Z","abstract_excerpt":"We prove the ill-posedness for the 3D incompressible inhomogeneous Navier-stokes equations in critical Besov space. In particular, a norm inflation happens in finite time with the initial data satisfying $$\\|a_0\\|_{\\dot{B}_{p,1}^\\frac{3}{p}}+\\|u_0\\|_{\\dot{B}_{6,1}^{-\\frac{1}{2}}}\\le \\delta,\\ p>6$$ or $$\\|a_0\\|_{\\dot{B}_{6,1}^\\frac{1}{2}}+\\|u_0\\|_{\\dot{B}_{p,1}^{\\frac{3}{p}-1}}\\le \\delta,\\ p>6.$$ To obtain the norm inflation, we construct a special class of initial data and introduce a modified pressure. Comparing with the classical Navier-Stokes equations in $L^\\infty$ framework, we can obtain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04551","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"}