{"paper":{"title":"The Navier--Stokes equations in exterior Lipschitz domains: $\\mathrm{L}^p$-theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Keiichi Watanabe, Patrick Tolksdorf","submitted_at":"2019-06-06T17:30:28Z","abstract_excerpt":"We show that the Stokes operator defined on $\\mathrm{L}^p_{\\sigma} (\\Omega)$ for an exterior Lipschitz domain $\\Omega \\subset \\mathbb{R}^n$ $(n \\geq 3)$ admits maximal regularity provided that $p$ satisfies $| 1/p - 1/2| < 1/(2n) + \\varepsilon$ for some $\\varepsilon > 0$. In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on $\\mathrm{L}^p_\\sigma (\\Omega)$ for such $p$. In addition, $\\mathrm{L}^p$-$\\mathrm{L}^q$-mapping properties of the Stokes semigroup and its gradient with optimal decay estimates are obtained. This enables us to prove the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.02713","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"}