{"paper":{"title":"Eulerian dynamics with a commutator forcing II: flocking","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"E. Tadmor, R. Shvydkoy","submitted_at":"2017-01-26T14:11:32Z","abstract_excerpt":"We continue our study of one-dimensional class of Euler equations, introduced in \\cite{ST2016}, driven by a forcing with a commutator structure of the form $[\\aL_\\phi,u](\\rho)=\\phi*(\\rho u)- (\\phi*\\rho)u$, where $u$ is the velocity field and $\\phi$ belongs to a rather general class of \\emph{influence} or interaction kernels.\n  In this paper we quantify the large-time behavior of such systems in terms of \\emph{fast flocking} for two prototypical sub-classes of kernels: bounded positive $\\phi$'s, and singular $\\phi(r) = r^{-(1+\\a)}$ of order $\\alpha\\in [1,2)$ associated with the action of the fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.07710","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"}