{"paper":{"title":"On some free boundary problem of the Navier-Stokes equations in the maximal $L_p$-$L_q$ regularity class","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yoshihiro Shibata","submitted_at":"2015-01-09T07:08:06Z","abstract_excerpt":"This paper is concerned with the free boundary problem for the Navier Stokes equations without surface tension in the $L_p$ in time and $L_q$ in space setting with $2 < p < \\infty$ and $N < q < \\infty$. A local in time existence theorem is proved in a uniform $W^{2-1/q}_q$ domain in the $N$-dimensional Euclidean space ${\\Bbb R}^N$ ($N \\geq 2$) under the assumption that weak Dirichlet-Neumann problem is uniquely solvable. Moreover, a global in time existence theorem is proved for small initial data under the assumption that $\\Omega$ is bounded additionally. This was already proved by Solonnikov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02054","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"}