{"paper":{"title":"Decay of the solution to the bipolar Euler-Poisson system with damping in $\\mathbb{R}^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Weike Wang, Zhigang Wu","submitted_at":"2012-12-16T06:19:40Z","abstract_excerpt":"We construct the global solution to the Cauchy's problem of the bipolar Euler-Poisson equations with damping in $\\mathbb{R}^3$ when $H^3$ norm of the initial data is small. If further, the $\\dot{H}^{-s}$ norm ($0\\leq s<3/2)$ or $\\dot{B}_{2,\\infty}^{-s}$ norm ($0<s\\leq3/2$) of the initial data is bounded, we give the optimal decay rates of the solution. As a byproduct, the decay results of the $L^p-L^2$ ($1\\leq p\\leq2$) type hold without the smallness of the $L^p$ norm of the initial data. In particular, we deduce that $\\|\\nabla^k(\\rho_1-\\rho_2)\\|_{L^2} \\sim(1+t)^{-5/4-\\frac{k}{2}}$ and $\\|\\nab"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3754","kind":"arxiv","version":2},"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"}