{"paper":{"title":"Large Time Decay Estimates for the Muskat Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Neel Patel, Robert M. Strain","submitted_at":"2016-10-17T19:11:25Z","abstract_excerpt":"We prove time decay of solutions to the Muskat equation in 2D and in 3D. In \\cite{JEMS} and \\cite{CCGRPS}, the authors introduce the norms $\\|f\\|_{s}(t)= \\int_{\\mathbb{R}^{2}} |\\xi|^{s}|\\hat{f}(\\xi)| \\ d\\xi$ in order to prove global existence of solutions to the Muskat problem. In this paper, for the 3D Muskat problem, given initial data $f_{0}\\in H^{l}(\\mathbb{R}^{2})$ for some $l\\geq 3$ such that $\\|f_{0}\\|_{1} < k_{0}$ for a constant $k_{0} \\approx 1/5$, we prove uniform in time bounds of $\\|f\\|_{s}(t)$ for $-d < s < l-1$ and assuming $\\|f_{0}\\|_{\\nu} < \\infty$ we prove time decay estimates"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05271","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"}