{"paper":{"title":"Mixing and Un-mixing by Incompressible Flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrej Zlatos, Yao Yao","submitted_at":"2014-07-15T22:20:16Z","abstract_excerpt":"We consider the questions of efficient mixing and un-mixing by incompressible flows which satisfy periodic, no-flow, or no-slip boundary conditions on a square. Under the uniform-in-time constraint $\\|\\nabla u(\\cdot,t)\\|_p\\leq 1$ we show that any function can be mixed to scale $\\epsilon$ in time $O(|\\log\\epsilon|^{1+\\nu_p})$, with $\\nu_p=0$ for $p<\\tfrac{3+\\sqrt 5}2$ and $\\nu_p\\leq \\tfrac 13$ for $p\\geq \\tfrac{3+\\sqrt 5}2$. Known lower bounds show that this rate is optimal for $p\\in(1,\\tfrac{3+\\sqrt 5}2)$. We also show that any set which is mixed to scale $\\epsilon$ but not much more than that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.4163","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"}