{"paper":{"title":"Convergence Rates to Stationary Solutions of a Gas-liquid Model with External Forces and Vacuum","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changjiang Zhu, Long Fan, Qingqing liu","submitted_at":"2011-10-28T04:48:05Z","abstract_excerpt":"In this paper, we study the asymptotic behavior of solutions to a Gas-liquid model with external forces and general pressure law. Under some suitable assumptions on the initial date and $\\gamma>1$, if $\\theta\\in(0,\\frac{\\gamma}{2}]\\cap(0,\\gamma-1]\\cap(0,1-\\alpha\\gamma]$, we prove the weak solution $(cQ(x,t),u(x,t))$ behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. In addition, if $\\theta\\in(0,\\frac{\\gamma}{2}]\\cap(0,\\gamma-1)\\cap(0,1-\\alpha\\gamma]$, following the same idea in \\cite{Fang-Zhang4}, we estimate the stabilization rate of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6248","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"}