{"paper":{"title":"A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Genevi\\`eve Raugel (LM-Orsay), Romain Joly (IF)","submitted_at":"2010-05-11T06:46:42Z","abstract_excerpt":"The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\\dot y(t)=g(y(t))$ on $\\mathbb{R}^d$ and those of the parabolic equations $\\dot u=\\Delta u +f(x,u,\\nabla u)$ on a bounded domain $\\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\\geq 3$ and in the corresponding cases $\\Omega=(0,1)$, $\\Omega=\\mathbb{T}^1$ and dim($\\Omega$)$\\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1727","kind":"arxiv","version":4},"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"}