{"paper":{"title":"Unstable manifolds of Euler equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chongchun Zeng, Zhiwu Lin","submitted_at":"2011-12-19T22:59:36Z","abstract_excerpt":"We consider a steady state $v_{0}$ of the Euler equation in a fixed bounded domain in $\\mathbf{R}^{n}$. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler equation as an ODE on an infinite dimensional manifold of volume preserving maps in $W^{k, q}$, $(k>1+\\frac{n}{q})$, the unstable (and stable) manifolds of $v_{0}$ are constructed under certain spectral gap condition which is verified for both 2D and 3D examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instabili"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4525","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"}